SUppose T: ℝ3→ℝ2 is a linear transformation. Three vectors U1, U2 and U3 are given below together with their images by T. Find T(W) for the given vector W.

Question

U1 = <1,−1,2> U2 = <−3,3,−6> U3 = <2,1,5> W = <−7,−2,−17> T(U1) = <−15,2> T(U2) = <45,−6> T(U3) = <−20,−6> T(W) = ???

Answer 1

Let the 3×3 matrix U = (u₁,u₂,u₃) have the given 3-vectors in its columns.

We need to solve the system U(a,b,c)⸆ = w for the 3-vector (a,b,c)⸆ e.g., by row reduction of the 3×4 matrix (U,w).

It turns out U is degenerate, so (a,b,c) = (a,(1 + a)/3,–3) for any a value.

Then T(w) = T(U(a,(1 + a)/3,–3)⸆) = T(U)(a,(1 + a)/3,–3)⸆ = (Tu₁,Tu₂,Tu₃)(a,(1 + a)/3,–3)⸆ = (75,16)⸆ for any a value.