In: Math
Find a matrix representation of transformation T(x)= 2x1w1+x2w2-3x3w3 from R3 to a vector space W, where w1,w2, and w3 ∈ W. Clearly state how this matrix is representing the transformation.
T:R3 → W is defined by T(X) = T(x1,x2,x3)T = 2x1 w1+x2 w2-3x3 w3, where w1,w2, and w3 ∈ W.
Then T(e1) = T(1,0,0)T = 2w1+0w2+0w3, T(e2) = T(0,1,0)T = 0w1+1w2+0w3 and T(e3) = T(0,0,1)T = 0w1+0w2-3w3. Hence, the standard matrix of T is A =
2w1 |
0 |
0 |
0 |
w2 |
0 |
0 |
0 |
-3w3 |
Then T(X) = AX.
Let X =(x1,x2,x3)T and Y=(y1,y2,y3)T be 2 arbitrary vectors in R3 and let k be an arbitrary scalar. Then T(X+Y) = A(X+Y) = AX +AY = T(X) +T(Y). This implies that T preserves vector addition.
Also, T(kX)= A(kX) = kAX = kT(X). This implies that T preserves scalar multiplication also.
Thus, Tis a linear transformation.
If we need actual multiplication by A, then T(X+Y) = A(X+Y) = A(x1+y1,x2+y2,x3+y3)T = 2(x1+y1)w1+(x2+y2)w2-3 (x3+y3)w3 = 2x1w1+x2w2-3x33+2y1w1+y2w2-3y3w3 = A(x1,x2,x3)T+A(y1,y2,y3)T = AX+AY = T(X)+T(Y). This implies that T preserves vector addition.
Also, T(kX) = A(kX) = A(kx1k,x2,kx3)T = 2kx1w1+kx2w2-3kx w3 = k(2x1 w1+x2 w2-3x3 w3,) = kAX = kT(X). This implies that T preserves scalar multiplication also.
Hence , Tis a linear transformation.