Question

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.

Answer 1

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(kx_1k,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.