T;R4>R3 be linear such that T(1,0,0,0)=(2,3,6) T(0,1,0,0)=(1,2,0) T(0,0,1,0)=(1,2,-3) T(0,0,0,1)=(0,2,-1) find the matrix of T in the basis std basis

Question

its a tuitorial

Answer 1

T is a 3-row, 4-column matrix, mapping 4-space into 3-space:

⎡ 2  1  1  0 ⎤

⎪ 3  2  2  2 ⎥

⎣ 6  0 -3 -1 ⎦

When applied to vector (1,0,0,0) we get:

⎡ 2  1  1  0 ⎤⎡1 ⎤  ⎡2⎤

⎪ 3  2  2  2 ⎥⎪0 ⎪=⎪3⎪

⎣ 6  0 -3 -1 ⎦⎪0 ⎪  ⎣6⎦

                         ⎣0 ⎦

Note that column 1 is duplicated in the product, and this applies similarly to the other vectors, duplicating successive columns in T. Whichever row in the vector contains 1 causes the corresponding column in T to be duplicated in the product.