Linear Transformation matrix help

In summary, the conversation discusses finding a nonzero 2x2 matrix A such that Ax is parallel to the vector [1] [2] for all x in R2. Two methods are suggested, one using a vector representation and the other using a matrix representation. The goal is to find values for A that satisfy the given conditions.
  • #1
cwatki14
57
0
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.
 
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  • #2
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?
 
  • #3
What Mark44 suggests works fine. A more "primitive method" is to write A as
[tex]\begin{bmatrix}a & b \\ c & d\end{bmatrix}[/tex]
so your equation "Ax= k[1 2]T" becomes
[tex]\begin{bmatrix}a & b \\ c & d\end{bmatrix}[/tex]\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.
 
  • #4
Note: fixed your LaTeX by adding a missing [ tex] tag.
HallsofIvy said:
What Mark44 suggests works fine. A more "primitive method" is to write A as
[tex]\begin{bmatrix}a & b \\ c & d\end{bmatrix}[/tex]
so your equation "Ax= k[1 2]T" becomes
[tex]\begin{bmatrix}a & b \\ c & d\end{bmatrix}[/tex] [tex]\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.
That's what I meant by saying "specify values for A." I think we're on the same page here.
 

1. What is a linear transformation matrix?

A linear transformation matrix is a mathematical representation of a linear transformation, which is a function that maps one vector space to another in a way that preserves the vector space structure.

2. How do you create a linear transformation matrix?

To create a linear transformation matrix, you first need to define the transformation and its corresponding vector spaces. Then, you can represent the transformation as a matrix by assigning each input vector to a column in the matrix and each output vector to a row.

3. What is the purpose of a linear transformation matrix?

The purpose of a linear transformation matrix is to simplify and generalize the process of performing linear transformations. By representing a transformation as a matrix, it becomes easier to manipulate and apply to different vectors or vector spaces.

4. How do you apply a linear transformation matrix to a vector?

To apply a linear transformation matrix to a vector, you simply multiply the vector by the matrix. The resulting vector will be the transformed version of the original vector according to the transformation defined by the matrix.

5. Can a linear transformation matrix be used for non-linear transformations?

No, a linear transformation matrix can only be used for linear transformations. Non-linear transformations require more complex mathematical representations, such as polynomial functions or neural networks.

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