Question

(1 point) Let $v_1 = \begin{bmatrix} 1 \\ 4 \end{bmatrix}$ and $v_2 = \begin{bmatrix} 1 \\ 3 \end{bmatrix}$. Let $T: \mathbb{R}^2 \to \mathbb{R}^2$ be the linear transformation satisfying $T(v_1) = \begin{bmatrix} 31 \\ -11 \end{bmatrix}$ and $T(v_2) = \begin{bmatrix} 25 \\ -6 \end{bmatrix}$. Find the image of an arbitrary vector $\begin{bmatrix} x \\ y \end{bmatrix}$. $T \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} \\ \end{bmatrix}$.

Name: 1 point let v1 beginbmatrix 1 4 endbmatrix and v2 beginbmatrix 1 3 endbmatrix let t mathbbr2 to mathbbr2 be the linear transformation satisfying tv1 beginbmatrix 31 11 endbmatrix and tv2 19218
Uploaded: 2024-03-04T14:12:52-08:00
Duration: 2 min 19 s
Channel: Shu-Ting Huang
Description: 1 point let v1 beginbmatrix 1 4 endbmatrix and v2 beginbmatrix 1 3 endbmatrix let t mathbbr2 to mathbbr2 be the linear transformation satisfying tv1 beginbmatrix 31 11 endbmatrix and tv2 19218

          (1 point)
Let $v_1 = \begin{bmatrix} 1 \\ 4 \end{bmatrix}$ and $v_2 = \begin{bmatrix} 1 \\ 3 \end{bmatrix}$.
Let $T: \mathbb{R}^2 \to \mathbb{R}^2$ be the linear transformation satisfying
$T(v_1) = \begin{bmatrix} 31 \\ -11 \end{bmatrix}$ and $T(v_2) = \begin{bmatrix} 25 \\ -6 \end{bmatrix}$.
Find the image of an arbitrary vector $\begin{bmatrix} x \\ y \end{bmatrix}$.
$T \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix}  \\  \end{bmatrix}$.

1 point let v1 beginbmatrix 1 4 endbmatrix and v2 beginbmatrix 1 3 endbmatrix let t mathbbr2 to mathbbr2 be the linear transformation satisfying tv1 beginbmatrix 31 11 endbmatrix and tv2 19218

Added by Michael A.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Shu-Ting Huang

03/04/2024

Step 1

That is, we want to find scalars $c_1$ and $c_2$ such that $\begin{bmatrix} x \\ y \end{bmatrix} = c_1 v_1 + c_2 v_2 = c_1 \begin{bmatrix} 1 \\ 4 \end{bmatrix} + c_2 \begin{bmatrix} 1 \\ 3 \end{bmatrix} = \begin{bmatrix} c_1 + c_2 \\ 4c_1 + 3c_2 Show more…

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(1 point) Let $v_1 = \begin{bmatrix} 1 \\ 4 \end{bmatrix}$ and $v_2 = \begin{bmatrix} 1 \\ 3 \end{bmatrix}$. Let $T: \mathbb{R}^2 \to \mathbb{R}^2$ be the linear transformation satisfying $T(v_1) = \begin{bmatrix} 31 \\ -11 \end{bmatrix}$ and $T(v_2) = \begin{bmatrix} 25 \\ -6 \end{bmatrix}$. Find the image of an arbitrary vector $\begin{bmatrix} x \\ y \end{bmatrix}$. $T \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} \\ \end{bmatrix}$.