MathsGee Q&A - Recent questions tagged space
https://mathsgee.com/tag/space
Powered by Question2AnswerIn circle \(O, R Q \perp P M, O Q=Q N\) and \(x=\frac{M R}{M P}\). Calculate \(x\)
https://mathsgee.com/39381/in-circle-o-r-q-perp-p-m-o-q-q-n-and-x-frac-m-r-m-p-calculate-x
<p>In circle \(O, R Q \perp P M, O Q=Q N\) and \(x=\frac{M R}{M P}\). Calculate \(x\)</p>
<p><img alt="calculate" src="https://mathsgee.com/?qa=blob&qa_blobid=12693673454619497759" style="height:202px; width:265px"></p>Mathematicshttps://mathsgee.com/39381/in-circle-o-r-q-perp-p-m-o-q-q-n-and-x-frac-m-r-m-p-calculate-xSun, 17 Apr 2022 08:17:28 +0000Two mobile network companies run a customer retention (churn reduction) program. Given the results below, which campaign was more successful. Please explain your answer.
https://mathsgee.com/38820/companies-customer-retention-reduction-campaign-successful
<p>Two mobile network companies run a customer retention (churn reduction) program. Given the results below, which campaign was more successful. Please explain your answer.</p>
<p><img alt="churn modelling" src="https://mathsgee.com/?qa=blob&qa_blobid=11019936572847742472" style="height:709px; width:459px"></p>
<p>source: <a href="https://stochasticsolutions.com/pdf/SavedAndDrivenAway.pdf" rel="nofollow" target="_blank">https://stochasticsolutions.com/pdf/SavedAndDrivenAway.pdf</a></p>
<p> </p>
<p> </p>Data Science & Statisticshttps://mathsgee.com/38820/companies-customer-retention-reduction-campaign-successfulSun, 20 Mar 2022 04:47:29 +0000How have data-driven customer churn/retention approaches evolved over time in the telecommunications space for mobile network operators?
https://mathsgee.com/38808/customer-retention-approaches-telecommunications-operators
How have data-driven customer churn/retention approaches evolved over time in the telecommunications space for mobile network operators?Advertising & Marketinghttps://mathsgee.com/38808/customer-retention-approaches-telecommunications-operatorsSun, 20 Mar 2022 04:06:48 +0000When Earth is seen from outer space, it looks mainly blue. This is because most of the earth is covered with ______
https://mathsgee.com/38703/earth-outer-space-looks-mainly-because-earth-covered
When Earth is seen from outer space, it looks mainly blue. This is because most of the earth is covered with ______Physics & Chemistryhttps://mathsgee.com/38703/earth-outer-space-looks-mainly-because-earth-coveredTue, 15 Mar 2022 18:34:46 +0000Which measurement is more accurate: taking Earthâ€™s surface temperature from the ground or from space?
https://mathsgee.com/37871/measurement-accurate-taking-earths-surface-temperature-ground
Which measurement is more accurate: taking Earth&rsquo;s surface temperature from the ground or from space?Geography & Environmenthttps://mathsgee.com/37871/measurement-accurate-taking-earths-surface-temperature-groundSat, 19 Feb 2022 01:51:57 +0000When is a metric useful?
https://mathsgee.com/37746/when-is-a-metric-useful
When is a metric useful?Advertising & Marketinghttps://mathsgee.com/37746/when-is-a-metric-usefulTue, 15 Feb 2022 23:41:05 +0000What Is a Metric?
https://mathsgee.com/37730/what-is-a-metric
<p>What Is a Metric?</p>
<p><img alt="business metrics" src="https://mathsgee.com/?qa=blob&qa_blobid=17129979075198945653" style="height:450px; width:600px"></p>Advertising & Marketinghttps://mathsgee.com/37730/what-is-a-metricTue, 15 Feb 2022 23:13:57 +0000A fair coin is tossed, and a fair die is thrown. Write down sample spaces for
https://mathsgee.com/37561/fair-coin-tossed-and-fair-die-thrown-write-down-sample-spaces
A fair coin is tossed, and a fair die is thrown. Write down sample spaces for<br />
<br />
(a) the toss of the coin;<br />
(b) the throw of the die;<br />
(c) the combination of these experiments.<br />
<br />
Let \(\mathrm{A}\) be the event that a head is tossed, and \(\mathrm{B}\) be the event that an odd number is thrown. Directly from the sample space, calculate \(\mathrm{P}(A \cap B)\) and \(\mathrm{P}(A \cup B)\).Data Science & Statisticshttps://mathsgee.com/37561/fair-coin-tossed-and-fair-die-thrown-write-down-sample-spacesTue, 08 Feb 2022 13:53:36 +0000Describe the sample space and all 16 events for a trial in which two coins are thrown and each shows either a head or a tail.
https://mathsgee.com/37559/describe-sample-space-events-trial-which-thrown-shows-either
Describe the sample space and all 16 events for a trial in which two coins are thrown and each shows either a head or a tail.Data Science & Statisticshttps://mathsgee.com/37559/describe-sample-space-events-trial-which-thrown-shows-eitherTue, 08 Feb 2022 13:52:26 +0000Calculate \(P(500)\) (the enrollment in Math 151 in fall of 2500). Simplify your answer as much as possible. The answer will be quite large.
https://mathsgee.com/37312/calculate-enrollment-simplify-answer-possible-answer-quite
Assume that Math 151 in fall of 2000 had an enrollment of 500 students and in fall 2002 had an enrollment of 750 students. Assume also that if \(P(t)\) is the enrollment at time \(t\) (let \(t\) be in years, with \(t=0\) corresponding to year 2000), then \(P^{\prime}(t)=k P(t)\) for some constant \(k\). Calculate \(P(500)\) (the enrollment in Math 151 in fall of 2500). Simplify your answer as much as possible. The answer will be quite large.Mathematicshttps://mathsgee.com/37312/calculate-enrollment-simplify-answer-possible-answer-quiteFri, 04 Feb 2022 09:43:09 +0000Let \(v_{1} \ldots v_{k}\) be vectors in a linear space with an inner product \(\langle,\),\(rangle . Define the\) Gram determinant by \(G\left(v_{1}, \ldots, v_{k}\right)=\operatorname{det}\left(\left\langle v_{i}, v_{j}\right\rangle\right)\).
https://mathsgee.com/36470/vectors-product-define-determinant-operatorname-langle-rangle
Let \(v_{1} \ldots v_{k}\) be vectors in a linear space with an inner product \(\langle,\),\(rangle . Define the\) Gram determinant by \(G\left(v_{1}, \ldots, v_{k}\right)=\operatorname{det}\left(\left\langle v_{i}, v_{j}\right\rangle\right)\).<br />
a) If the \(v_{1} \ldots v_{k}\) are orthogonal, compute their Gram determinant.<br />
b) Show that the \(v_{1} \ldots v_{k}\) are linearly independent if and only if their Gram determinant is not zero.<br />
c) Better yet, if the \(v_{1} \ldots v_{k}\) are linearly independent, show that the symmetric matrix \(\left(\left\langle v_{i}, v_{j}\right\rangle\right)\) is positive definite. In particular, the inequality \(G\left(v_{1}, v_{2}\right) \geq 0\) is the Schwarz inequality.<br />
d) Conversely, if \(A\) is any \(n \times n\) positive definite matrix, show that there are vectors \(v_{1}, \ldots, v_{n}\) so that \(A=\left(\left\langle v_{i}, v_{j}\right\rangle\right)\).<br />
e) Let \(\mathcal{S}\) denote the subspace spanned by the linearly independent vectors \(w_{1} \ldots w_{k} .\) If \(X\) is any vector, let \(P_{\mathcal{S}} X\) be the orthogonal projection of \(X\) into \(\mathcal{S}\), prove that the distance \(\left\|X-P_{\mathcal{S}} X\right\|\) from \(X\) to \(\mathcal{S}\) is given by the formula<br />
\[<br />
\left\|X-Z_{\mathcal{S}} X\right\|^{2}=\frac{G\left(X, w_{1}, \ldots, w_{k}\right)}{G\left(w_{1}, \ldots, w_{k}\right)} .<br />
\]Mathematicshttps://mathsgee.com/36470/vectors-product-define-determinant-operatorname-langle-rangleFri, 21 Jan 2022 08:46:45 +0000Let \(A\) be a positive definite \(n \times n\) real matrix, \(\vec{b}\) a real vector, and \(\vec{N}\) a real unit vector.
https://mathsgee.com/36467/positive-definite-times-matrix-real-vector-real-unit-vector
Let \(A\) be a positive definite \(n \times n\) real matrix, \(\vec{b}\) a real vector, and \(\vec{N}\) a real unit vector.<br />
a) For which value(s) of the real scalar \(c\) is the set<br />
\[<br />
E:=\left\{\vec{x} \in \mathbb{R}^{3} \mid\langle\vec{x}, A \vec{x}\rangle+2\langle\vec{b}, \vec{x}\rangle+c=0\right\}<br />
\]<br />
(an ellipsoid) non-empty? <br />
<br />
b) For what value(s) of the scalar \(d\) is the plane \(Z:=\left\{\vec{x} \in \mathbb{R}^{3} \mid\langle\vec{N}, \vec{x}\rangle=d\right\}\) tangent to the above ellipsoid \(E\) (assumed non-empty)?<br />
<br />
[SUGGESTION: First discuss the case where \(A=I\) and \(\vec{b}=0\). Then show how by a change of variables, the general case can be reduced to this special case. ]Mathematicshttps://mathsgee.com/36467/positive-definite-times-matrix-real-vector-real-unit-vectorFri, 21 Jan 2022 08:41:48 +0000Let \(\vec{x}\) and \(\vec{p}\) be points in \(\mathbb{R}^{n}\). Under what conditions on the scalar \(c\) is the set \[ \|\vec{x}\|^{2}+2\langle\vec{p}, \vec{x}\rangle+c=0 \]
https://mathsgee.com/36466/points-mathbb-under-what-conditions-the-scalar-langle-rangle
a) Let \(\vec{x}\) and \(\vec{p}\) be points in \(\mathbb{R}^{n}\). Under what conditions on the scalar \(c\) is the set<br />
\[<br />
\|\vec{x}\|^{2}+2\langle\vec{p}, \vec{x}\rangle+c=0<br />
\]<br />
a sphere \(\left\|\vec{x}-\vec{x}_{0}\right\|=R \geq 0\) ? Compute the center, \(\vec{x}_{0}\), and radius, \(R\), in terms of \(\vec{p}\) and \(c\).<br />
b) Let<br />
\[<br />
\begin{aligned}<br />
Q(\vec{x}) &=\sum a_{i j} x_{i} x_{j}+2 \sum b_{i} x_{i}+c \\<br />
&=\langle\vec{x}, A \vec{x}\rangle+2\langle\vec{b}, \vec{x}\rangle+c<br />
\end{aligned}<br />
\]<br />
be a real quadratic polynomial so \(\vec{x}=\left(x_{1}, \ldots, x_{n}\right), \vec{b}=\left(b_{1}, \ldots, b_{n}\right)\) are real vectors and \(A=\left(a_{i j}\right)\) is a real symmetric \(n \times n\) matrix. Just as in the case \(n=1\) (which you should do first), if \(A\) is invertible find a vector \(\vec{v}\) (depending on \(A\) and \(\vec{b}\) ) so that the change of variables \(\vec{y}==\vec{x}-\vec{v}\) (this is a translation by the vector \(\vec{v}\) ) so that in the new \(\vec{y}\) variables \(Q\) has the simpler form<br />
\[<br />
Q=\langle\vec{y}, A \vec{y}\rangle+\gamma \text { that is, } Q=\sum a_{i j} y_{i} y_{j}+\gamma,<br />
\]<br />
where \(\gamma=c-\left\langle\vec{b}, A^{-1} \vec{b}\right\rangle\).<br />
As an example, apply this to \(Q(\vec{x})=2 x_{1}^{2}+2 x_{1} x_{2}+3 x_{2}-4\).Mathematicshttps://mathsgee.com/36466/points-mathbb-under-what-conditions-the-scalar-langle-rangleFri, 21 Jan 2022 08:39:47 +0000Let \(V \subset \mathbb{R}^{n}\) be a linear space, \(Q: R^{n} \rightarrow V^{\perp}\) the orthogonal projection into \(V^{\perp}\), and \(x \in \mathbb{R}^{n}\) a given vector.
https://mathsgee.com/36465/subset-mathbb-rightarrow-orthogonal-projection-mathbb-vector
[Dual variational problems] Let \(V \subset \mathbb{R}^{n}\) be a linear space, \(Q: R^{n} \rightarrow V^{\perp}\) the orthogonal projection into \(V^{\perp}\), and \(x \in \mathbb{R}^{n}\) a given vector. Note that \(Q=I-P\), where \(P\) in the orthogonal projection into \(V\)<br />
a) Show that \(\max _{\{z \perp V,\|z\|=1\}}\langle x, z\rangle=\|Q x\|\).<br />
b) Show that \(\min _{v \in V}\|x-v\|=\|Q x\|\).<br />
[Remark: dual variational problems are a pair of maximum and minimum problems whose extremal values are equal.]Mathematicshttps://mathsgee.com/36465/subset-mathbb-rightarrow-orthogonal-projection-mathbb-vectorFri, 21 Jan 2022 08:38:58 +0000Let \(C[-1,1]\) be the real inner product space consisting of all continuous functions \(f:[-1,1] \rightarrow \mathbb{R}\), with the inner product \(\langle f, g\rangle:=\int_{-1}^{1} f(x) g(x) d x\).
https://mathsgee.com/36462/product-consisting-continuous-functions-rightarrow-product
Let \(C[-1,1]\) be the real inner product space consisting of all continuous functions \(f:[-1,1] \rightarrow \mathbb{R}\), with the inner product \(\langle f, g\rangle:=\int_{-1}^{1} f(x) g(x) d x\). Let \(W\) be the subspace of odd functions, i.e. functions satisfying \(f(-x)=-f(x)\). Find (with proof) the orthogonal complement of \(W\).Mathematicshttps://mathsgee.com/36462/product-consisting-continuous-functions-rightarrow-productFri, 21 Jan 2022 08:34:34 +0000Let \(\mathcal{P}_{2}\) be the space of polynomials \(p(x)=a+b x+c x^{2}\) of degree at most 2 with the inner product \(\langle p, q\rangle=\int_{-1}^{1} p(x) q(x) d x\).
https://mathsgee.com/36461/mathcal-space-polynomials-degree-inner-product-langle-rangle
Let \(\mathcal{P}_{2}\) be the space of polynomials \(p(x)=a+b x+c x^{2}\) of degree at most 2 with the inner product \(\langle p, q\rangle=\int_{-1}^{1} p(x) q(x) d x\). Let \(\ell\) be the functional \(\ell(p):=p(0)\). Find \(h \in \mathcal{P}_{2}\) so that \(\ell(p)=\langle h, p\rangle\) for all \(p \in \mathcal{P}_{2}\).Mathematicshttps://mathsgee.com/36461/mathcal-space-polynomials-degree-inner-product-langle-rangleFri, 21 Jan 2022 08:33:56 +0000Let \(\mathcal{P}_{2}\) be the space of quadratic polynomials.
https://mathsgee.com/36460/let-mathcal-p-2-be-the-space-of-quadratic-polynomials
Let \(\mathcal{P}_{2}\) be the space of quadratic polynomials.<br />
<br />
a) Show that \(\langle f, g\rangle=f(-1) g(-1)+f(0) g(0)+f(1) g(1)\) is an inner product for this space.<br />
<br />
b) Using this inner product, find an orthonormal basis for \(\mathcal{P}_{2}\).<br />
<br />
c) Is this also an inner product for the space \(\mathcal{P}_{3}\) of polynomials of degree at most three? Why?Mathematicshttps://mathsgee.com/36460/let-mathcal-p-2-be-the-space-of-quadratic-polynomialsFri, 21 Jan 2022 08:33:18 +0000In a complex vector space (with a hermitian inner product), if a matrix \(A\) satisfies \(\langle X, A X\rangle=0\) for all vectors \(X\), show that \(A=0\). [The previous problem shows that this is false in a real vector space].
https://mathsgee.com/36456/complex-hermitian-product-satisfies-vectors-previous-problem
In a complex vector space (with a hermitian inner product), if a matrix \(A\) satisfies \(\langle X, A X\rangle=0\) for all vectors \(X\), show that \(A=0\). [The previous problem shows that this is false in a real vector space].Mathematicshttps://mathsgee.com/36456/complex-hermitian-product-satisfies-vectors-previous-problemFri, 21 Jan 2022 08:27:16 +0000Let \(A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}\) be a linear map defined by the matrix \(A\).
https://mathsgee.com/36449/let-mathbb-rightarrow-mathbb-linear-map-defined-the-matrix
Let \(A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}\) be a linear map defined by the matrix \(A\). If the matrix \(B\) satisfies the relation \(\langle A X, Y\rangle=\langle X, B Y\rangle\) for all vectors \(X \in \mathbb{R}^{n}, Y \in \mathbb{R}^{k}\), show that \(B\) is the transpose of \(A\), so \(B=A^{T}\). [This basic property of the transpose,<br />
\[<br />
\langle A X, Y\rangle=\left\langle X, A^{T} Y\right\rangle,<br />
\]<br />
is the only reason the transpose is important.]Mathematicshttps://mathsgee.com/36449/let-mathbb-rightarrow-mathbb-linear-map-defined-the-matrixFri, 21 Jan 2022 08:17:51 +0000Let \(A\) be a positive definite \(n \times n\) real matrix, \(b \in \mathbb{R}^{n}\), and consider the quadratic polynomial \[ Q(x):=\frac{1}{2}\langle x, A x\rangle-\langle b, x\rangle \]
https://mathsgee.com/36446/positive-definite-mathbb-consider-quadratic-polynomial-langle
Let \(A\) be a positive definite \(n \times n\) real matrix, \(b \in \mathbb{R}^{n}\), and consider the quadratic polynomial<br />
\[<br />
Q(x):=\frac{1}{2}\langle x, A x\rangle-\langle b, x\rangle<br />
\]<br />
a) Show that \(Q\) is bounded below, that is, there is a constant \(m\) so that \(Q(x) \geq m\) for all \(x \in \mathbb{R}^{n}\).<br />
b) Show that \(Q\) blows up at infinity by showing that there are positive constants \(R\) and \(c\) so that if \(\|x\| \geq R\), then \(Q(x) \geq c\|x\|^{2}\).<br />
33<br />
c) If \(x_{0} \in \mathbb{R}^{n}\) minimizes \(Q\), show that \(A x_{0}=b\). [Moral: One way to solve \(A x=b\) is to minimize \(Q .\) ]<br />
d) Give an example showing that if \(A\) is only positive semi-definite, then \(Q(x)\) may not be bounded below.Mathematicshttps://mathsgee.com/36446/positive-definite-mathbb-consider-quadratic-polynomial-langleFri, 21 Jan 2022 08:15:07 +0000Let \(\ell\) be any linear functional. Show there is a unique vector \(v \in \mathbb{R}^{n}\) so that \(\ell(x):=\langle x, v\rangle\).
https://mathsgee.com/36445/linear-functional-there-unique-vector-mathbb-langle-rangle
[LINEAR FUNCTIONALS] In \(R^{n}\) with the usual inner product, a linear functional \(\ell:\) \(\mathbb{R}^{n} \rightarrow \mathbb{R}\) is just a linear map into the reals (in a complex vector space, it maps into the complex numbers \(\mathbb{C}\) ). Define the norm, \(\|\ell\|\), as<br />
\[<br />
\|\ell\|:=\max _{\|x\|=1}|\ell(x)| .<br />
\]<br />
a) Show that the set of linear functionals with this norm is a normed linear space.<br />
b) If \(v \in \mathbb{R}^{n}\) is a given vector, define \(\ell(x)=\langle x, v\rangle\). Show that \(\ell\) is a linear functional and that \(\|\ell\|=\|v\|\).<br />
c) [REPRESENTATION OF A LINEAR FUNCTIONAL] Let \(\ell\) be any linear functional. Show there is a unique vector \(v \in \mathbb{R}^{n}\) so that \(\ell(x):=\langle x, v\rangle\).<br />
d) [EXTENSION OF A LINEAR FUNCTIONAL] Let \(U \subset \mathbb{R}^{n}\) be a subspace of \(\mathbb{R}^{n}\) and \(\ell\) a linear functional defined on \(U\) with norm \(\|\ell\|_{U}\). Show there is a unique extension of \(\ell\) to \(\mathbb{R}^{n}\) with the property that \(\|\ell\|_{\mathbb{R}^{n}}=\|\ell\|_{U}\).<br />
[In other words define \(\ell\) on all of \(\mathbb{R}^{n}\) so that on \(U\) this extended definition agrees with the original definition and so that its norm is unchanged].Mathematicshttps://mathsgee.com/36445/linear-functional-there-unique-vector-mathbb-langle-rangleFri, 21 Jan 2022 08:14:22 +0000Let \(w(x)\) be a positive continuous function on the interval \(0 \leq x \leq 1, n\) a positive integer, and \(\mathcal{P}_{n}\) the vector space of polynomials \(p(x)\) whose degrees are at most \(n\) equipped with the inner product
https://mathsgee.com/36444/positive-continuous-function-interval-polynomials-equipped
Let \(w(x)\) be a positive continuous function on the interval \(0 \leq x \leq 1, n\) a positive integer, and \(\mathcal{P}_{n}\) the vector space of polynomials \(p(x)\) whose degrees are at most \(n\) equipped with the inner product<br />
\[<br />
\langle p, q\rangle=\int_{0}^{1} p(x) q(x) w(x) d x .<br />
\]<br />
a) Prove that \(\mathcal{P}_{n}\) has an orthonormal basis \(p_{0}, p_{1}, \ldots, p_{n}\) with the degree of \(p_{k}\) is \(k\) for each \(k\).<br />
b) Prove that \(\left\langle p_{k}, p_{k}^{\prime}\right\rangle=0\) for each \(k\).Mathematicshttps://mathsgee.com/36444/positive-continuous-function-interval-polynomials-equippedFri, 21 Jan 2022 08:12:34 +0000Let \(V\) be the real vector space of continuous real-valued functions on the closed interval \([0,1]\), and let \(w \in V\). For \(p, q \in V\), define \(\langle p, q\rangle=\int_{0}^{1} p(x) q(x) w(x) d x\).
https://mathsgee.com/36443/vector-continuous-functions-interval-define-langle-rangle
Let \(V\) be the real vector space of continuous real-valued functions on the closed interval \([0,1]\), and let \(w \in V\). For \(p, q \in V\), define \(\langle p, q\rangle=\int_{0}^{1} p(x) q(x) w(x) d x\).<br />
a) Suppose that \(w(a)>0\) for all \(a \in[0,1]\). Does it follow that the above defines an inner product on \(V\) ? Justify your assertion.<br />
<br />
b) Does there exist a choice of \(w\) such that \(w(1 / 2)<0\) and such that the above defines an inner product on \(V ?\) Justify your assertion.Mathematicshttps://mathsgee.com/36443/vector-continuous-functions-interval-define-langle-rangleFri, 21 Jan 2022 08:11:57 +0000Let \(V, W\) be vectors in \(\mathbb{R}^{n}\).
https://mathsgee.com/36435/let-v-w-be-vectors-in-mathbb-r-n
Let \(V, W\) be vectors in \(\mathbb{R}^{n}\).<br />
<br />
a) Show that the Pythagorean relation \(\|V+W\|^{2}=\|V\|^{2}+\|W\|^{2}\) holds if and only if \(V\) and \(W\) are orthogonal.<br />
<br />
b) Prove the parallelogram identity \(\|V+W\|^{2}+\|V-W\|^{2}=2\|V\|^{2}+2\|W\|^{2}\) and interpret it geometrically. [This is true in any real inner product space].Mathematicshttps://mathsgee.com/36435/let-v-w-be-vectors-in-mathbb-r-nFri, 21 Jan 2022 08:04:40 +0000Let \(\vec{e}_{1}, \vec{e}_{2}\), and \(\vec{e}_{3}\) be the standard basis for \(\mathbb{R}^{3}\) and let \(L: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) be a linear transformation with the properties
https://mathsgee.com/36416/standard-rightarrow-mathbb-linear-transformation-properties
Let \(\vec{e}_{1}, \vec{e}_{2}\), and \(\vec{e}_{3}\) be the standard basis for \(\mathbb{R}^{3}\) and let \(L: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) be a linear transformation with the properties<br />
\[<br />
L\left(\vec{e}_{1}\right)=\vec{e}_{2}, \quad L\left(\vec{e}_{2}\right)=2 \vec{e}_{1}+\vec{e}_{2}, \quad L\left(\vec{e}_{1}+\vec{e}_{2}+\vec{e}_{3}\right)=\vec{e}_{3} .<br />
\]<br />
Find a vector \(\vec{v}\) such that \(L(\vec{v})=k \vec{v}\) for some real number \(k\).Mathematicshttps://mathsgee.com/36416/standard-rightarrow-mathbb-linear-transformation-propertiesFri, 21 Jan 2022 02:28:17 +0000Let \(U \subset V\) and \(W\) be finite dimensional linear spaces and \(L: V \rightarrow W\) a linear map. Show that
https://mathsgee.com/36392/subset-finite-dimensional-linear-spaces-rightarrow-linear
Let \(U \subset V\) and \(W\) be finite dimensional linear spaces and \(L: V \rightarrow W\) a linear map. Show that<br />
\[<br />
\operatorname{dim}\left(\left.\operatorname{ker} L\right|_{U}\right) \leq \operatorname{dim} \operatorname{ker} L=\operatorname{dim} V-\operatorname{dim} \operatorname{Im}(L)<br />
\]Mathematicshttps://mathsgee.com/36392/subset-finite-dimensional-linear-spaces-rightarrow-linearFri, 21 Jan 2022 01:49:15 +0000Let \(A: \mathbb{R}^{\ell} \rightarrow \mathbb{R}^{n}\) and \(B: \mathbb{R}^{k} \rightarrow \mathbb{R}^{\ell}\).
https://mathsgee.com/36391/let-mathbb-rightarrow-mathbb-and-mathbb-rightarrow-mathbb
Let \(A: \mathbb{R}^{\ell} \rightarrow \mathbb{R}^{n}\) and \(B: \mathbb{R}^{k} \rightarrow \mathbb{R}^{\ell}\). Prove that \(\operatorname{rank} A+\operatorname{rank} B-\ell \leq \operatorname{rank} A B \leq \min \{\operatorname{rank} A, \operatorname{rank} B\} .\)<br />
\(\left[\right.\) HINT: Observe that \(\left.\operatorname{rank}(A B)=\left.\operatorname{rank} A\right|_{\operatorname{Image}(B)} \cdot\right]\)Mathematicshttps://mathsgee.com/36391/let-mathbb-rightarrow-mathbb-and-mathbb-rightarrow-mathbbFri, 21 Jan 2022 01:48:36 +0000Let \(L: V \rightarrow V\) be a linear map on a vector space \(V\).
https://mathsgee.com/36382/let-l-v-rightarrow-v-be-a-linear-map-on-a-vector-space-v
Let \(L: V \rightarrow V\) be a linear map on a vector space \(V\).<br />
a) Show that \(\operatorname{ker} L \subset \operatorname{ker} L^{2}\) and, more generally, \(\operatorname{ker} L^{k} \subset \operatorname{ker} L^{k+1}\) for all \(k \geq 1\).<br />
b) If \(\operatorname{ker} L^{j}=\operatorname{ker} L^{j+1}\) for some integer \(j\), show that \(\operatorname{ker} L^{k}=\operatorname{ker} L^{k+1}\) for all \(k \geq j\). Does your proof require that \(V\) is finite dimensional?<br />
c) Let \(A\) be an \(n \times n\) matrix. If \(A^{j}=0\) for some integer \(j\) (perhaps \(j>n\) ), show that \(A^{n}=0\).Mathematicshttps://mathsgee.com/36382/let-l-v-rightarrow-v-be-a-linear-map-on-a-vector-space-vFri, 21 Jan 2022 01:41:34 +0000Which of the following are not a basis for the vector space of all symmetric \(2 \times 2\) matrices? Why?
https://mathsgee.com/36379/which-following-basis-vector-space-symmetric-times-matrices
Which of the following are not a basis for the vector space of all symmetric \(2 \times 2\) matrices? Why?<br />
<br />
<br />
<br />
a) \(\left(\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right),\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right),\left(\begin{array}{ll}0 & 0 \\ 0 & 1\end{array}\right)\)<br />
b) \(\left(\begin{array}{ll}3 & 3 \\ 3 & 3\end{array}\right),\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right),\left(\begin{array}{ll}1 & 1 \\ 1 & 0\end{array}\right)\)<br />
c) \(\left(\begin{array}{ll}1 & 1 \\ 1 & 0\end{array}\right),\left(\begin{array}{cc}1 & 2 \\ 2 & -3\end{array}\right),\left(\begin{array}{ll}0 & 1 \\ 1 & 1\end{array}\right)\)<br />
d) \(\left(\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right),\left(\begin{array}{ll}1 & 1 \\ 1 & 0\end{array}\right),\left(\begin{array}{cc}-2 & -2 \\ -2 & 1\end{array}\right)\)<br />
e) \(\left(\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right),\left(\begin{array}{ll}-1 & -1 \\ -1 & -1\end{array}\right),\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)\)<br />
f) \(\left(\begin{array}{ll}1 & 0 \\ 0 & 0\end{array}\right),\left(\begin{array}{cc}-1 & 2 \\ 2 & -1\end{array}\right),\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)\)Mathematicshttps://mathsgee.com/36379/which-following-basis-vector-space-symmetric-times-matricesFri, 21 Jan 2022 01:39:34 +0000Let \(\mathcal{S}\) be the linear space of infinite sequences of real numbers \(x:=\left(x_{1}, x_{2}, \ldots\right) .\) Define the linear map \(L: \mathcal{S} \rightarrow \mathcal{S}\) by
https://mathsgee.com/36369/mathcal-infinite-sequences-numbers-mathcal-rightarrow-mathcal
Let \(\mathcal{S}\) be the linear space of infinite sequences of real numbers \(x:=\left(x_{1}, x_{2}, \ldots\right) .\) Define the linear map \(L: \mathcal{S} \rightarrow \mathcal{S}\) by<br />
\[<br />
L x:=\left(x_{1}+x_{2}, x_{2}+x_{3}, x_{3}+x_{4}, \ldots\right) .<br />
\]<br />
a) Find a basis for the nullspace of \(L\). What is its dimension?<br />
b) What is the image of \(L ?\) Justify your assertion.<br />
c) Compute the eigenvalues of \(L\) and an eigenvector corresponding to each eigenvalue.Mathematicshttps://mathsgee.com/36369/mathcal-infinite-sequences-numbers-mathcal-rightarrow-mathcalFri, 21 Jan 2022 01:29:55 +0000Give an example of a linear transformation \(L: V \rightarrow V\) (or show that there is no such transformation) for which:
https://mathsgee.com/36368/example-linear-transformation-rightarrow-transformation
Let \(V\) be a vector space with \(\operatorname{dim} V=10\) and let \(L: V \rightarrow V\) be a linear transformation. Consider \(L^{k}: V \rightarrow V, k=1,2,3, \ldots\) Let \(r_{k}=\operatorname{dim}\left(\operatorname{Im} L^{k}\right)\), that is, \(r_{k}\) is the dimension of the image of \(L^{k}, k=1,2, \ldots\)<br />
<br />
Give an example of a linear transformation \(L: V \rightarrow V\) (or show that there is no such transformation) for which:<br />
a) \(\left(r_{1}, r_{2}, \ldots\right)=(10,9, \ldots)\)<br />
b) \(\left(r_{1}, r_{2}, \ldots\right)=(8,5, \ldots)\);<br />
c) \(\left(r_{1}, r_{2}, \ldots\right)=(8,6,4,4, \ldots)\).Mathematicshttps://mathsgee.com/36368/example-linear-transformation-rightarrow-transformationFri, 21 Jan 2022 01:29:06 +0000Given a unit vector \(\mathbf{w} \in \mathbb{R}^{n}\), let \(W=\operatorname{span}\{\mathbf{w}\}\) and consider the linear map \(T: \mathbb{R}^{n} \rightarrow\) \(\mathbb{R}^{n}\) defined by
https://mathsgee.com/36366/operatorname-mathbf-consider-linear-mathbb-rightarrow-defined
Given a unit vector \(\mathbf{w} \in \mathbb{R}^{n}\), let \(W=\operatorname{span}\{\mathbf{w}\}\) and consider the linear map \(T: \mathbb{R}^{n} \rightarrow\) \(\mathbb{R}^{n}\) defined by<br />
\[<br />
T(\mathbf{x})=2 \operatorname{Proj}_{W}(\mathbf{x})-\mathbf{x},<br />
\]<br />
where \(\operatorname{Proj}_{W}(\mathbf{x})\) is the orthogonal projection onto \(W\). Show that \(T\) is one-to-one.Mathematicshttps://mathsgee.com/36366/operatorname-mathbf-consider-linear-mathbb-rightarrow-definedFri, 21 Jan 2022 01:26:46 +0000Find the dimension of \(A\) considered as a real vector space.
https://mathsgee.com/36364/find-the-dimension-of-a-considered-as-a-real-vector-space
Consider the two linear transformations on the vector space \(V=\mathbf{R}^{n}\) :<br />
\(R=\) right shift: \(\left(x_{1}, \ldots, x_{n}\right) \rightarrow\left(0, x_{1}, \ldots, x_{n-1}\right)\)<br />
\(L=\) left shift: \(\left(x_{1}, \ldots, x_{n}\right) \rightarrow\left(x_{2}, \ldots, x_{n}, 0\right)\)<br />
Let \(A \subset\) End \((V)\) be the real algebra generated by \(\mathrm{R}\) and \(\mathrm{L}\). Find the dimension of \(A\) considered as a real vector space.Mathematicshttps://mathsgee.com/36364/find-the-dimension-of-a-considered-as-a-real-vector-spaceFri, 21 Jan 2022 01:24:54 +0000Let \(\mathcal{P}_{2}\) be the space of polynomials of degree at most 2 .
https://mathsgee.com/36360/let-mathcal-p-2-be-the-space-of-polynomials-of-degree-at-most
Let \(\mathcal{P}_{2}\) be the space of polynomials of degree at most 2 .<br />
<br />
<br />
<br />
a) Find a basis for this space.<br />
b) Let \(D: \mathcal{P}_{2} \rightarrow \mathcal{P}_{2}\) be the derivative operator \(D=d / d x\). Using the basis you picked in the previous part, write \(D\) as a matrix. Compute \(D^{3}\) in this situation. Why should you have predicted this without computation?Mathematicshttps://mathsgee.com/36360/let-mathcal-p-2-be-the-space-of-polynomials-of-degree-at-mostFri, 21 Jan 2022 01:20:47 +0000Show that \(\mathcal{L}\) and \(\mathcal{R}\) are linear spaces and compute their dimensions.
https://mathsgee.com/36357/show-mathcal-mathcal-linear-spaces-compute-their-dimensions
Say \(A \in M(n, \mathbb{F})\) has rank \(k\). Define<br />
\[<br />
\mathcal{L}:=\{B \in M(n, \mathbb{F}) \mid B A=0\} \quad \text { and } \quad \mathcal{R}:=\{C \in M(n, \mathbb{F}) \mid A C=0\} .<br />
\]<br />
Show that \(\mathcal{L}\) and \(\mathcal{R}\) are linear spaces and compute their dimensions.Mathematicshttps://mathsgee.com/36357/show-mathcal-mathcal-linear-spaces-compute-their-dimensionsFri, 21 Jan 2022 01:18:42 +0000Let \(\mathcal{P}_{3}\) be the linear space of polynomials \(p(x)\) of degree at most 3 .
https://mathsgee.com/36356/let-mathcal-the-linear-space-of-polynomials-of-degree-at-most
Let \(\mathcal{P}_{3}\) be the linear space of polynomials \(p(x)\) of degree at most 3 . Give a non-trivial example of a linear map \(L: \mathcal{P}_{3} \rightarrow \mathcal{P}_{3}\) that is nilpotent, that is, \(L^{k}=0\) for some integer \(k\). [A trivial example is the zero map: \(L=0 .\) ]Mathematicshttps://mathsgee.com/36356/let-mathcal-the-linear-space-of-polynomials-of-degree-at-mostFri, 21 Jan 2022 01:18:05 +0000Show that \(\mathcal{A}\) is a linear space and compute its dimension.
https://mathsgee.com/36349/show-that-mathcal-is-linear-space-and-compute-its-dimension
Let \(V \subset \mathbb{R}^{11}\) be a linear subspace of dimension 4 and consider the family \(\mathcal{A}\) of all linear maps \(L: \mathbb{R}^{11}->\mathbb{R}^{9}\) each of whose nullspace contain \(V\).<br />
Show that \(\mathcal{A}\) is a linear space and compute its dimension.Mathematicshttps://mathsgee.com/36349/show-that-mathcal-is-linear-space-and-compute-its-dimensionFri, 21 Jan 2022 01:13:07 +0000Let \(V, W\) be two-dimensional real vector spaces, and let \(f_{1}, \ldots, f_{5}\) be linear transformations from \(V\) to \(W\).
https://mathsgee.com/36348/dimensional-vector-spaces-ldots-linear-transformations-from
Let \(V, W\) be two-dimensional real vector spaces, and let \(f_{1}, \ldots, f_{5}\) be linear transformations from \(V\) to \(W\). Show that there exist real numbers \(a_{1}, \ldots, a_{5}\), not all zero, such that \(a_{1} f_{1}+\cdots+a_{5} f_{5}\) is the zero transformation.Mathematicshttps://mathsgee.com/36348/dimensional-vector-spaces-ldots-linear-transformations-fromFri, 21 Jan 2022 01:12:16 +0000Let \(\mathcal{M}_{(3,2)}\) be the linear space of all \(3 \times 2\) real matrices and let the linear map \(L\) : \(\mathcal{M}_{(3,2)} \rightarrow \mathbb{R}^{5}\) be onto. Compute the dimension of the nullspace of \(L\).
https://mathsgee.com/36342/mathcal-matrices-rightarrow-compute-dimension-nullspace
Let \(\mathcal{M}_{(3,2)}\) be the linear space of all \(3 \times 2\) real matrices and let the linear map \(L\) : \(\mathcal{M}_{(3,2)} \rightarrow \mathbb{R}^{5}\) be onto. Compute the dimension of the nullspace of \(L\).Mathematicshttps://mathsgee.com/36342/mathcal-matrices-rightarrow-compute-dimension-nullspaceFri, 21 Jan 2022 01:05:12 +0000Let \(V\) be a vector space and \(\ell: V \rightarrow \mathbb{R}\) be a linear map.
https://mathsgee.com/36334/let-v-be-vector-space-and-ell-rightarrow-mathbb-be-linear-map
Let \(V\) be a vector space and \(\ell: V \rightarrow \mathbb{R}\) be a linear map. If \(z \in V\) is not in the nullspace of \(\ell\), show that every \(x \in V\) can be decomposed uniquely as \(x=v+c z\), where \(v\) is in the nullspace of \(\ell\) and \(c\) is a scalar. [MORAL: The nullspace of a linear functional has codimension one.]Mathematicshttps://mathsgee.com/36334/let-v-be-vector-space-and-ell-rightarrow-mathbb-be-linear-mapFri, 21 Jan 2022 00:57:56 +0000Identify which of the following collections of matrices form a linear subspace in the linear space \(\operatorname{Mat}_{2 \times 2}(\mathbb{R})\) of all \(2 \times 2\) real matrices?
https://mathsgee.com/36332/identify-following-collections-matrices-subspace-operatorname
Identify which of the following collections of matrices form a linear subspace in the linear space \(\operatorname{Mat}_{2 \times 2}(\mathbb{R})\) of all \(2 \times 2\) real matrices?<br />
<br />
a) All invertible matrices.<br />
b) All matrices that satisfy \(A^{2}=0\).<br />
c) All anti-symmetric matrices, that is, \(A^{T}=-A\).<br />
d) Let \(B\) be a fixed matrix and \(\mathcal{B}\) the set of matrices with the property that \(A^{T} B=\) \(B A^{T}\).Mathematicshttps://mathsgee.com/36332/identify-following-collections-matrices-subspace-operatornameFri, 21 Jan 2022 00:55:59 +0000Call a subset \(S\) of a vector space \(V\) a spanning set if \(\operatorname{Span}(S)=V\). Suppose that \(T: V \rightarrow W\) is a linear map of vector spaces.
https://mathsgee.com/36307/subset-spanning-operatorname-suppose-rightarrow-vector-spaces
Call a subset \(S\) of a vector space \(V\) a spanning set if \(\operatorname{Span}(S)=V\). Suppose that \(T: V \rightarrow W\) is a linear map of vector spaces.<br />
a) Prove that a linear map \(T\) is 1-1 if and only if \(T\) sends linearly independent sets to linearly independent sets.<br />
b) Prove that \(T\) is onto if and only if \(T\) sends spanning sets to spanning sets.Mathematicshttps://mathsgee.com/36307/subset-spanning-operatorname-suppose-rightarrow-vector-spacesThu, 20 Jan 2022 12:59:58 +0000Does an 8-dimensional vector space contain linear subspaces \(V_{1}, V_{2}, V_{3}\) with no common non-zero element, such that
https://mathsgee.com/36304/dimensional-vector-contain-linear-subspaces-common-element
Does an 8-dimensional vector space contain linear subspaces \(V_{1}, V_{2}, V_{3}\) with no common non-zero element, such that<br />
<br />
<br />
<br />
a). \(\operatorname{dim}\left(V_{i}\right)=5, i=1,2,3\) ?<br />
b). \(\operatorname{dim}\left(V_{i}\right)=6, \quad i=1,2,3 ?\)Mathematicshttps://mathsgee.com/36304/dimensional-vector-contain-linear-subspaces-common-elementThu, 20 Jan 2022 12:52:43 +0000Define the Discrete-Time Fourier time/space convolution property
https://mathsgee.com/35350/define-discrete-time-fourier-time-space-convolution-property
Define the Discrete-Time Fourier time/space convolution propertyMathematicshttps://mathsgee.com/35350/define-discrete-time-fourier-time-space-convolution-propertyTue, 11 Jan 2022 13:40:11 +0000What is the difference between a 2-dimensional and 3-dimensional space?
https://mathsgee.com/34806/what-the-difference-between-dimensional-dimensional-space
What is the difference between a 2-dimensional and 3-dimensional space?Mathematicshttps://mathsgee.com/34806/what-the-difference-between-dimensional-dimensional-spaceMon, 03 Jan 2022 06:48:09 +0000If a 12 inch diameter pizza requires 10 ounces of dough, how much dough is needed for a 16 inch pizza?
https://mathsgee.com/34794/diameter-pizza-requires-ounces-dough-much-dough-needed-pizza
If a 12 inch diameter pizza requires 10 ounces of dough, how much dough is needed for a 16 inch pizza?Mathematicshttps://mathsgee.com/34794/diameter-pizza-requires-ounces-dough-much-dough-needed-pizzaMon, 03 Jan 2022 01:19:28 +0000What is a sample space when calculating probability?
https://mathsgee.com/34633/what-is-a-sample-space-when-calculating-probability
What is a sample space when calculating probability?Data Science & Statisticshttps://mathsgee.com/34633/what-is-a-sample-space-when-calculating-probabilityWed, 15 Dec 2021 18:31:32 +0000What is a clinical trial?
https://mathsgee.com/33928/what-is-a-clinical-trial
What is a clinical trial?Data Science & Statisticshttps://mathsgee.com/33928/what-is-a-clinical-trialSat, 16 Oct 2021 05:26:43 +0000Which of the following scientists is responsible for the exclusion principle which states that two objects may NOT occupy the same space at the same time? Was it:
https://mathsgee.com/33290/following-scientists-responsible-exclusion-principle-objects
Which of the following scientists is responsible for the exclusion principle which states that two objects may NOT occupy the same space at the same time? Was it:<br />
<br />
w) Heisenberg<br />
<br />
x) Bohr<br />
<br />
y) Teller<br />
<br />
z) PauliPhysics & Chemistryhttps://mathsgee.com/33290/following-scientists-responsible-exclusion-principle-objectsTue, 05 Oct 2021 03:22:17 +0000What is an electric field at a point?
https://mathsgee.com/33233/what-is-an-electric-field-at-a-point
What is an electric field at a point?Physics & Chemistryhttps://mathsgee.com/33233/what-is-an-electric-field-at-a-pointFri, 01 Oct 2021 00:35:37 +0000