Linear Transformation matrix help

[cwatki14]

The problem is as follows:
Find a nonzero 2x2 matrix A such that Ax is parallel to the vector
[1]
[2]
for all x in R².

So far I know A=[v1 v2] therefore Ax= [v1 v2][x₁]
[x₂]
= x₁v₁+x₂v₂
I know these two vectors are parallel, but I am a little stuck how to relate this property to solve for the matrix A.

 

[Mark44]

You have Ax = k[1 2]^T for any vector x in R².
Specify values for A.

What does A do to [1 0]^T? to [0 1]^T?

 

[HallsofIvy]

What Mark44 suggests works fine. A more "primitive method" is to write A as
$$\begin{bmatrix}a & b \\ c & d\end{bmatrix}$$
so your equation "Ax= k[1 2]^T" becomes
$$\begin{bmatrix}a & b \\ c & d\end{bmatrix}$$\begin{bmatrix} x \\ y\end{bmatrix}= \begin{bmatrix}ax+ by \\ cx+ dy\end{bmatrix}= \begin{matrix}k \\ 2k\end{bmatrix}[/tex]

giving you two equations, ax+ by= k and cx+ dy= 2k for the 5 unknown numbers. There will, of course, be an infinite number of possible answers. You are simply asked to find one such matrix.

 

[Mark44]

Note: fixed your LaTeX by adding a missing [ tex] tag.

What Mark44 suggests works fine. A more "primitive method" is to write A as
$$\begin{bmatrix}a & b \\ c & d\end{bmatrix}$$
so your equation "Ax= k[1 2]^T" becomes
$$\begin{bmatrix}a & b \\ c & d\end{bmatrix}$$ $$\begin{bmatrix} x \\ y\end{bmatrix}= \begin{bmatrix}ax+ by \\ cx+ dy\end{bmatrix}= \begin{matrix}k \\ 2k\end{bmatrix}$$

giving you two equations, ax+ by= k and cx+ dy= 2k for the 5 unknown numbers. There will, of course, be an infinite number of possible answers. You are simply asked to find one such matrix.

That's what I meant by saying "specify values for A." I think we're on the same page here.