Suppose that T: R^3 rightarrow R^2 is a linear transformation and that T ([1 2 1

Question

Suppose that T: R^3 rightarrow R^2 is a linear transformation and that T ([1 2 1]) = [-1 2] and T([2 - 1 1]) = [3 0]. Compute T (2[2 -1 1]), T ([3 6 3]), and T ([-1 3 0]). Suppose that T: R^3 rightarrow R^2 is defined by T([x_1 x_2 x_3]) = [x_1 + 2x_2 + x_3 3x_1 - x_2 - x_3. Find Suppose T: R^2 rightarrow R^2 is a linear transformation. In each case use the information provide to find the standard matrix A for T. T([1 0]) = [2 -3]) and T([2 1]) = [-1 1] T ([2 1]) = [5 3] and T ([0 1]) = [1 -3] T([1 1]) = [3 3] and T ([1 -1]) = [-1 1] determine whether each of the following functions is a linear transformation provide a proof; if not explain why. T ([x_1 x_2) = [x_1 + 2x_2 x^2_2] T([x_1 x_2]) = [x_1 + 2x_2 0]

Explanation / Answer

3(b) We have T(2,1)T = (5,3)T and T(0,1)T = (1,-3)T.

Now, (1,0)T = ½{(2,1)T-(0,1)T}. Hence T (1,0)T = T [½{(2,1)T-(0,1)T}] = ½ T[(2,1)T-(0,1)T] = ½[ T(2,1)T- T(0,1)T] = ½[ (5,3)T- (1,-3)T] = ½ ( 4,6)T = (2,3)T.

If A is the standard matrix for T , the the 1st and the 2nd columns of A are T (1,0)T and T(0,1)T.   Thus, A =

2

1

3

-3

2

1

3

-3

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