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## Homework Statement

Say if f is a linear transformation from R

^{2}to R

^{3}with f(1,0) = (1,2,3) and f(0,1) = (0,-1,2).

Determine f(x,y).

## The Attempt at a Solution

I understand the theorem on linear transformation and bases but unsure as to how I should apply it in practice. Should I be performing the linear transformation test? But the question has already specified that f is a linear transformation.

Edit: {u1,u2,u3...un} is a basis for R

^{n}and {t1,t2,t3...tn} is a basis for R

^{m}

then there is a unique linear transformation such that f maps (u1) to t1: f(u1) = t1

This can be expressed as f(u1) = t1, f(u2) = t2, f(u3) = t3...f(un) = tn

f:R

^{2}→R

^{3}

f(e1) = (1,2,3)

∴f(1,0) = (1,2,3)

∴f(1) = 1, f(0) = 2

f(e2) = (0,-1,2)

∴f(0,1) = (0,-1,2)

∴f(0) = 0, f(1) = -1

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