91Ó°ÊÓ

Let \(\beta\) and \(\gamma\) be the standard ordered bases for \(\mathrm{R}^{n}\) and \(\mathrm{R}^{m}\), respectively. For each linear transformation \(\mathrm{T}: \mathrm{R}^{n} \rightarrow \mathrm{R}^{m}\), compute \([\mathrm{T}]_{\beta}^{\gamma}\). (a) \(\mathrm{T}: \mathrm{R}^{2} \rightarrow \mathrm{R}^{3}\) defined by $\mathrm{T}\left(a_{1}, a_{2}\right)=\left(2 a_{1}-a_{2}, 3 a_{1}+4 a_{2}, a_{1}\right)$. (b) \(\mathrm{T}: \mathrm{R}^{3} \rightarrow \mathrm{R}^{2}\) defined by $\mathrm{T}\left(a_{1}, a_{2}, a_{3}\right)=\left(2 a_{1}+3 a_{2}-a_{3}, a_{1}+a_{3}\right)$. (c) \(\mathrm{T}: \mathrm{R}^{3} \rightarrow R\) defined by \(\mathrm{T}\left(a_{1}, a_{2}, a_{3}\right)=2 a_{1}+a_{2}-3 a_{3}\). (d) \(\mathrm{T}: \mathrm{R}^{3} \rightarrow \mathrm{R}^{3}\) defined by $$ \mathrm{T}\left(a_{1}, a_{2}, a_{3}\right)=\left(2 a_{2}+a_{3},-a_{1}+4 a_{2}+5 a_{3}, a_{1}+a_{3}\right) . $$ (e) \(\mathrm{T}: \mathrm{R}^{n} \rightarrow \mathrm{R}^{n}\) defined by $\mathrm{T}\left(a_{1}, a_{2}, \ldots, a_{n}\right)=\left(a_{1}, a_{1}, \ldots, a_{1}\right)$. (f) \(\mathrm{T}: \mathrm{R}^{n} \rightarrow \mathrm{R}^{n}\) defined by $\mathrm{T}\left(a_{1}, a_{2}, \ldots, a_{n}\right)=\left(a_{n}, a_{n-1}, \ldots, a_{1}\right)$. (g) \(\mathrm{T}: \mathrm{R}^{n} \rightarrow R\) defined by \(\mathrm{T}\left(a_{1}, a_{2}, \ldots, a_{n}\right)=a_{1}+a_{n}\).

Step by step solution

01

Apply T to basis elements of R^2

We are given the transformation T(a_1, a_2) = (2a_1 - a_2, 3a_1 + 4a_2, a_1). We apply T to the basis elements of R^2 which are e_1 = (1, 0) and e_2 = (0, 1). T(e_1) = T(1, 0) = (2, 3, 1) T(e_2) = T(0, 1) = (-1, 4, 0)

02

Write as linear combinations

Now, we express the resulting vectors as linear combinations of the standard basis for R^3: (1, 0, 0), (0, 1, 0) and (0, 0, 1). T(e_1) = 2(1, 0, 0) + 3(0, 1, 0) + 1(0, 0, 1) T(e_2) = -1(1, 0, 0) + 4(0, 1, 0) + 0(0, 0, 1)

03

Form the matrix representation [T]_β^γ

The coefficients of these linear combinations form the columns of the matrix representation [T]_β^γ. [T]_β^γ = \(\begin{bmatrix} 2 & -1 \\ 3 & 4 \\ 1 & 0 \end{bmatrix}\). (b)

04

Apply T to basis elements of R^3

We are given the transformation T(a_1, a_2, a_3) = (2a_1 + 3a_2 - a_3, a_1 + a_3). We apply T to the basis elements of R^3 which are e_1 = (1, 0, 0), e_2 = (0, 1, 0) and e_3 = (0, 0, 1). T(e_1) = T(1, 0, 0) = (2, 1) T(e_2) = T(0, 1, 0) = (3, 0) T(e_3) = T(0, 0, 1) = (-1, 1)

05

Write as linear combinations

Now, we express the resulting vectors as linear combinations of the standard basis for R^2: (1, 0) and (0, 1). T(e_1) = 2(1, 0) + 1(0, 1) T(e_2) = 3(1, 0) + 0(0, 1) T(e_3) = -1(1, 0) + 1(0, 1)

06

Form the matrix representation [T]_β^γ

The coefficients of these linear combinations form the columns of the matrix representation [T]_β^γ. [T]_β^γ = \(\begin{bmatrix} 2 & 3 & -1 \\ 1 & 0 & 1 \end{bmatrix}\). (c)

07

Apply T to basis elements of R^3

We are given the transformation T(a_1, a_2, a_3) = 2a_1 + a_2 - 3a_3. We apply T to the basis elements of R^3 which are e_1 = (1, 0, 0), e_2 = (0, 1, 0) and e_3 = (0, 0, 1). T(e_1) = T(1, 0, 0) = 2 T(e_2) = T(0, 1, 0) = 1 T(e_3) = T(0, 0, 1) = -3

08

Write as linear combinations

Now, we express the resulting scalars as linear combinations of the standard basis for R: (1). T(e_1) = 2(1) T(e_2) = 1(1) T(e_3) = -3(1)

09

Form the matrix representation [T]_β^γ

The coefficients of these linear combinations form the columns of the matrix representation [T]_β^γ. [T]_β^γ = \(\begin{bmatrix} 2 & 1 & -3 \end{bmatrix}\). Continue this process for parts (d), (e), (f), and (g) to compute the matrix representation [T]_β^γ for each given transformation.

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