How to Determine the Matrix of a Linear Map with a Non-Standard Basis?

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Homework Statement

T(2,1)---> (5,2) and T(1,2)--->(7,10) is a linear map on R^2. Determine the matrix T with respect to the basis B= {(3,3),(1,-1)}

Homework Equations

The Attempt at a Solution

matrix = 5 7
2 10 ?

 

[HallsofIvy]

Rather than the question marks, it would be better to show HOW you got that answer!

The simplest way to get the matrix representation for a linear transformation in a given basis is to apply the linear transformation to each basis vector in turn, writing the result in terms of the basis. The coefficients in the linear combination are the columns of the matrix.

Here, we know that T(2, 1)= (5, 2) and T(1, 2)= (7, 10). To determine what T does to (3, 3) and (1, -1), we need first to write them in terms of (2, 1) and (1, 2).
(3, 3)= a(2, 1)+ b(1, 2)= (2a+ b, a+ 2b) so we must have 2a+ b= 3 and a+ 2b= 3. Multiply the second equation by 2 and subtract fromthe first: (2a+ b)- 2(a+ 2b)= 2a+ b- 2a- 4b= -3b= 3 so b= -1. Then 2a- 1= 3 so 2a= 4 and a= 2.

T(3, 3)= T(2(2,1)- (1, 2))= 2T(2,1)- T(1,2)= 2(5,2)- (7, 10)= (10- 7, 4- 10)= (3, -6). Now we need to write that in terms of the basis (3, 3), (1, -1). Solve (3, -6)= x(3, 3)+ y(1, -1) for x and y. Then
$$\begin{array}x \\ y\end{array}$$
will be the first column of your matrix.

Do the same with (1, -1) to find the second column.