Find the matrix A such that T(x)=Ax
Let \(T: R^{3} \rightarrow R^{2}\) defined by \(T\left(\left[\begin{array}{l}x_{1} \\ x_{2} \\ x_{3}\end{array}\right]\right)=\left[\begin{array}{c}2 x_{1}+x_{3} \\ -x_{2}\end{array}\right]\).
a. Find the matrix \(A\) such that \(T(x)=A x\)
b. Demonstrate that \(T\) is a linear transformation.
Homework Answers
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linear algebra
Problem settingConsider the linear transformation \(\phi(\cdot): \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) on the standard vector space of dimension two over the field of real numbers defined as:$$ \phi\left(\left(\begin{array}{l} x_{0} \\ x_{1} \end{array}\right)\right)=\left(\begin{array}{r} 3 x_{0}-x_{1} \\ -7 x_{0}+2 x_{1} \end{array}\right) $$Problem taskFind \(\mathcal{R}_{G \rightarrow E}(\) id \()\) that is the change of basis matrix from basis \(G\) to the standard basis \(E\) where the standard basis vectors are:$$ \begin{array}{l} \vec{e}_{0}=\left(\begin{array}{l} 1 \\ 0 \end{array}\right) \\ \vec{e}_{1}=\left(\begin{array}{l} 0 \\ 1 \end{array}\right) \end{array} $$given that...
Help with Linear Algebra
1. Suppose that \(T\) is the matrix transformation defined by the matrix \(A\) and \(S\) the matrix transformation defined by \(B\) where$$ A=\left[\begin{array}{rrr} 3 & -1 & 0 \\ 1 & 2 & 2 \\ -1 & 3 & 2 \end{array}\right], \quad B=\left[\begin{array}{rrr} 1 & -1 & 0 \\ 2 & 1 & 2 \end{array}\right] $$a. If \(T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}\), what are the values of \(m\) and \(n ?\) What values of \(m\) and \(n\) are appropriate for the...
6. This problem uses least squares to find the curve y=ax+bx2 that best fits these 4...
This problem uses least squares to find the curve \(y=a x+b x^{2}\) that best fits these 4 points in the plane:$$ \left(x_{1}, y_{1}\right)=(-2,2), \quad\left(x_{2}, y_{2}\right)=(-1,1), \quad\left(x_{1}, y_{3}\right)=(1,0), \quad\left(x_{4}, y_{4}\right)=(2,2) . $$a. Write down 4 equations \(a x_{i}+b x_{i}^{2}=y_{i}, i=1,2,3,4\), that would be true if the line actually went through a11 four points.b. Now write those four equations in the form \(\mathbf{A}\left[\begin{array}{l}a \\ b\end{array}\right]=\mathbf{y}\)c. Now find \(\left[\begin{array}{l}\hat{a} \\ \hat{b}\end{array}\right]\) that minimizes \(\left\|A\left[\begin{array}{l}a \\ b\end{array}\right]-\mathbf{y}\right\|^{2}\).
Consider the 3 × 3 matrices
3. (3pts) Consider the \(3 \times 3\) matrices \(B=\left[\begin{array}{ccc}1 & 1 & 2 \\ -1 & 0 & 4 \\ 0 & 0 & 1\end{array}\right]\) and \(A=\left[\begin{array}{lll}\mathbf{a}_{1} & \mathbf{a}_{2} & \mathbf{a}_{3}\end{array}\right]\), where \(\mathbf{a}_{1}\), \(\mathbf{a}_{2}\), and \(\mathrm{a}_{9}\) are the columns of \(A\). Let \(A B=\left[\begin{array}{lll}v_{1} & v_{2} & v_{3}\end{array}\right]\), where \(v_{1}, v_{2}\), and \(v_{3}\) are the columns of the product. Express a as a linear combination of \(\mathbf{v}_{1}, \mathbf{v}_{2}\), and \(\mathbf{v}_{3}\).4. (3pts) Let \(T(x)=A x\) be the linear transformation given by$$...
Problem on Linear programming and Simplex method
Problem on Linear programming and Simplex methodThe \(\ell_{1}\) norm of a vector \(v \in \mathbb{R}\) is defined by$$ \|v\|_{1}:=\sum_{i=1}^{n}\left|v_{i}\right| $$Problems of the form Minimize \(\|v\|_{1}\) subject to \(v \in \mathbb{R}^{n}\) and \(A v=b\) arise very frequently in applied math, particularly in the field of compressed sensing.Consider the special case of this problem whith \(n=3\),$$ A=\left(\begin{array}{lll} 1 & 1 & 0 \\ 3 & 0 & 1 \end{array}\right) \quad \text { and } \quad b=\left(\begin{array}{l} 3 \\ 8 \end{array}\right) $$(a) (3...
could u help me for this question?thanku!! 21. Let T be a linear transformation from P2 into P3 over R defined by T(p(x)) xp(x). (a) Find [T]B.A the matrix of T relative to the bases A = {1-x, l-x...
could u help me for this question?thanku!!
21. Let T be a linear transformation from P2 into P3 over R defined by T(p(x)) xp(x). (a) Find [T]B.A the matrix of T relative to the bases A = {1-x, l-x2,x) and B={1,1+x, 1 +x+12, 1-x3}. (b) Use [TlB. A to find a basis for the range of T. (c) Use TB.A to find a basis for the kernel of T. (d) State the rank and nullity of T.
21. Let T...
Use A-1 to solve the following system of linear exuations
Let \(A=\left[\begin{array}{ccc}2 & 0 & -1 \\ 1 & -5 & 1 \\ 2 & -7 & 1\end{array}\right]\)a) Compute \(A^{-1} .\)b) Use \(A^{-1}\) to solve the following system of linear exuations:$$ \begin{array}{r} 2 x_{1}+-x_{3}=3 \\ x_{1}-5 x_{2}+x_{3}=1 \\ 2 x_{1}-7 x_{2}+x_{3}=4 \end{array} $$
Problem A A Markov chain Xo,X1,X2,... with state space 0,1,2) has the following transition matrix 0...
Problem A:A Markov chain \(X_{0}, X_{1}, X_{2}, \ldots\) with state space \{0,1,2\} has the following transition matrix$$ \boldsymbol{P}=\begin{array}{cccc} & 0 & 1 & 2 \\ 0 & 0.1 & 0.2 & a \\ 1 & 0.9 & 0.1 & 0 \\ 2 & 0.1 & 0.8 & b \end{array} $$and initial distribution \(\alpha_{0}=P\left(X_{0}=0\right)=0.3, \alpha_{1}=P\left(X_{0}=1\right)=0.4,\) and \(\alpha_{2}=P\left(X_{0}=2\right)=\)c. Find the following:a) values \(a, b\) and \(c\).b) \(P\left(X_{0}=0, X_{1}=1, X_{2}=2\right)\) and \(P\left(X_{0}=0, X_{1}=2, X_{2}=1\right)\).c) \(P\left(X_{1}=2\right)\)d) \(P\left(X_{2}=1, X_{3}=1 \mid X_{1}=2\right)\) and \(P\left(X_{1}=1, X_{2}=1 \mid...
Find the Trace of B
3. Let \(\quad B=\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right]\).(a) Find the Trace of B.(b) Find \(B^{-1}\), the inverse of \(B\).(c) A vector \(\vec{v}\) is an eigenvector of the matrix \(B\) if Matrix-Vector Multiplication \(B \vec{v}\) results in a scaling of the vector \(\vec{v}\). (i.e. \(B \vec{v}=c \vec{v}\), with \(c\) a real number.) Using the definition of Matrix-Vector Multiplication show that the vector \(\vec{v}=\left[\begin{array}{l}1 \\ 1\end{array}\right]\) is an eigenvector of \(B\) with eigenvalue \(c=3\).
QUESTION 4 Let T R3-P2 be defined by T(a, b, c) - (a + b + e) +(a+b)a2 (4.1) Show that T is a linear transformation (4.2) Fınd the matrix representation [T]s, B, of T relative to the basıs in R3 and...
QUESTION 4 Let T R3-P2 be defined by T(a, b, c) - (a + b + e) +(a+b)a2 (4.1) Show that T is a linear transformation (4.2) Fınd the matrix representation [T]s, B, of T relative to the basıs in R3 and the basis in P2, ordered from left to right Determine the range R(T of T Is T onto? In other words, is it true that R(T)P2 Let x, y E R3 Show that x-y ker(T) f and only...
3. (3pts) Consider the \(3 \times 3\) matrices \(B=\left[\begin{array}{ccc}1 & 1 & 2 \\ -1 & 0 & 4 \\ 0 & 0 & 1\end{array}\right]\) and \(A=\left[\begin{array}{lll}\mathbf{a}_{1} & \mathbf{a}_{2} & \mathbf{a}_{3}\end{array}\right]\), where \(\mathbf{a}_{1}\), \(\mathbf{a}_{2}\), and \(\mathrm{a}_{9}\) are the columns of \(A\). Let \(A B=\left[\begin{array}{lll}v_{1} & v_{2} & v_{3}\end{array}\right]\), where \(v_{1}, v_{2}\), and \(v_{3}\) are the columns of the product. Express a as a linear combination of \(\mathbf{v}_{1}, \mathbf{v}_{2}\), and \(\mathbf{v}_{3}\).4. (3pts) Let \(T(x)=A x\) be the linear transformation given by$$...
could u help me for this question?thanku!!
21. Let T be a linear transformation from P2 into P3 over R defined by T(p(x)) xp(x). (a) Find [T]B.A the matrix of T relative to the bases A = {1-x, l-x2,x) and B={1,1+x, 1 +x+12, 1-x3}. (b) Use [TlB. A to find a basis for the range of T. (c) Use TB.A to find a basis for the kernel of T. (d) State the rank and nullity of T.
21. Let T...
Problem A:A Markov chain \(X_{0}, X_{1}, X_{2}, \ldots\) with state space \{0,1,2\} has the following transition matrix$$ \boldsymbol{P}=\begin{array}{cccc} & 0 & 1 & 2 \\ 0 & 0.1 & 0.2 & a \\ 1 & 0.9 & 0.1 & 0 \\ 2 & 0.1 & 0.8 & b \end{array} $$and initial distribution \(\alpha_{0}=P\left(X_{0}=0\right)=0.3, \alpha_{1}=P\left(X_{0}=1\right)=0.4,\) and \(\alpha_{2}=P\left(X_{0}=2\right)=\)c. Find the following:a) values \(a, b\) and \(c\).b) \(P\left(X_{0}=0, X_{1}=1, X_{2}=2\right)\) and \(P\left(X_{0}=0, X_{1}=2, X_{2}=1\right)\).c) \(P\left(X_{1}=2\right)\)d) \(P\left(X_{2}=1, X_{3}=1 \mid X_{1}=2\right)\) and \(P\left(X_{1}=1, X_{2}=1 \mid...
3. Let \(\quad B=\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right]\).(a) Find the Trace of B.(b) Find \(B^{-1}\), the inverse of \(B\).(c) A vector \(\vec{v}\) is an eigenvector of the matrix \(B\) if Matrix-Vector Multiplication \(B \vec{v}\) results in a scaling of the vector \(\vec{v}\). (i.e. \(B \vec{v}=c \vec{v}\), with \(c\) a real number.) Using the definition of Matrix-Vector Multiplication show that the vector \(\vec{v}=\left[\begin{array}{l}1 \\ 1\end{array}\right]\) is an eigenvector of \(B\) with eigenvalue \(c=3\).
QUESTION 4 Let T R3-P2 be defined by T(a, b, c) - (a + b + e) +(a+b)a2 (4.1) Show that T is a linear transformation (4.2) Fınd the matrix representation [T]s, B, of T relative to the basıs in R3 and the basis in P2, ordered from left to right Determine the range R(T of T Is T onto? In other words, is it true that R(T)P2 Let x, y E R3 Show that x-y ker(T) f and only...
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