Thread: linear map question

If T : R^2 -----> R^2 is a linear map defined by T(a,b)=(a+b,a-b)

what is the matrix of T:

1) using the standard basis of R^2?

2) using the basis (-1,1),(0,-2) of R^2

Originally Posted by maximus101

If T : R^2 -----> R^2 is a linear map defined by T(a,b)=(a+b,a-b)

what is the matrix of T:

1) using the standard basis of R^2?

2) using the basis (-1,1),(0,-2) of R^2

Using the standard basis

$\displaystyle e_1=(1,0), e_2=(0,1)$ Then

$\displaystyle T(e_1)=(1,1)=e_1+e_2$ and
$\displaystyle T(e_2)=(1,-1)=e_1-e_2$

Remember that the transform of the basis vectors becomes the columns in the matrix representation.

This gives the standard matrix as

$\displaystyle \begin{bmatrix} 1 & 1 \\ 1 & -1\end{bmatrix}$

2 is exactly the same just use the basis given instead of the standard one.