Standard Matrix of Linear Transformation T: R2--R2

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In summary, The conversation discusses finding the standard matrix for a linear transformation T that switches the two coordinates of every point in a 2D vector space. The method suggested is to apply the transformation to the basis vectors and use the resulting coefficients as the columns of the matrix. The standard matrix is then multiplied by the basis vectors to get the coefficients.
  • #1
Derill03
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The question I am attempting to understand (my book absolutely sux) is this:

Give the standard matrix of the linear transformation T: R2--R2 which switches the two coordinates of every point?

Just not understanding what i am trying to accomplish here
 
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  • #2
Sorry about the notation.

Find a,b,c,d such that


| a b |
| c d | < x, y> = <y, x>
 
  • #3
Let us think about what matrix multiplication encodes. It is linear, so to understand what it does you just need to understand what it does to a basis (i.e. A(x+y)=Ax + Ay, so knowing what happens to x and y tells you what happens to x+y).

What happens if I multiply a matrix A into the column vector (1,0)^t? I get out the first column of A (check this!). Now find out what you get from multiplying into (0,1)^t

So if I know that A sends (1,0)^t to c and (0,1)^t to d, then I can write down A's matrix immediately.
 
  • #4
Here is the simplest way to find the matrix of a linear tranformation, in a given basis.

Apply the transformation to the first basis vector: write the result in terms of the basis. The coefficients are the first column in the matrix.

Now do the same to the second basis vector. That gives the second column.

I notice, now, that this is exactly what matt grime said!

That is, write the matrix, as you do:
[tex]\begin{bmatrix}a & b \\ c & d\end{bmatrix}[/tex]
and apply it to [1 0].
[tex]\begin{bmatrix}a & b \\ c & d\end{bmatrix}\begin{bmatrix}1 \\ 0\end{bmatrix}= \begin{bmatrix}a \\ c\end{bmatrix}[/tex]
But since this transformation "swaps the two coordinates", [1 0] is taken to [0 1] so you must have [a c]= [0 1].
 
  • #5
ok so my answer should be a 2x2 matrix?

I am to someway use the standard matrix
|1 0 |
|0 1 |
to get to my answer?

It seems that i should use the standard matrix muliplied by |0 1| to get a c, is this correct?
And then i could multiply standard matrix by |1 0| to get b d?
 
  • #6
Of course the matrix of the transformation T is a 2 x 2 matrix. It maps a vector in R2 to a different vector in R2.

Look at HallsOfIvy's post; he has done half the problem for you.
 

1. What is a Standard Matrix of Linear Transformation T?

A Standard Matrix of Linear Transformation T is a matrix representation of a linear transformation from a vector space to itself. In other words, it is a way to represent a function that maps a vector from one space to another.

2. How is a Standard Matrix of Linear Transformation T calculated?

The Standard Matrix of Linear Transformation T is calculated by applying the linear transformation to the standard basis vectors of the vector space. This will result in a matrix with the transformed vectors as its columns.

3. What is the purpose of a Standard Matrix of Linear Transformation T?

The purpose of a Standard Matrix of Linear Transformation T is to provide a way to easily perform calculations and analyze the properties of a linear transformation. It also allows for the transformation to be easily applied to other vectors in the vector space.

4. Can a Standard Matrix of Linear Transformation T represent any type of linear transformation?

Yes, a Standard Matrix of Linear Transformation T can represent any type of linear transformation as long as it is a transformation from a vector space to itself. This includes rotations, reflections, dilations, and combinations of these transformations.

5. How does the size of the Standard Matrix of Linear Transformation T relate to the dimension of the vector space?

The size of the Standard Matrix of Linear Transformation T will always be the same as the dimension of the vector space. For example, a linear transformation from R2 to R2 will have a 2x2 matrix representation, while a transformation from R3 to R3 will have a 3x3 matrix representation.

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