How can I use the given linear transformation to determine f(x,y)?

[negation]

Homework Statement

Say if f is a linear transformation from R² to R³ 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 Rⁿ 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:R² →R³

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

 

[HallsofIvy]

A linear transformation, f, has the property that f(\alpha u+ \beta v)= \alpha fu+ \beta fv where \alpha and \beta are scalars and u and v are vectors.

You know f(0,1) and f(1, 0) so given any (x, y), apply f to x(1, 0)+ y(0, 1).

 

[pasmith]

Homework Statement

Say if f is a linear transformation from R² to R³ 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.

Can you write (x,y) \in \mathbb{R}^2 as a linear combination of vectors for which you are given the value of f?

 

[negation]

A linear transformation, f, has the property that f(\alpha u+ \beta v)= \alpha fu+ \beta fv where \alpha and \beta are scalars and u and v are vectors.

You know f(0,1) and f(1, 0) so given any (x, y), apply f to x(1, 0)+ y(0, 1).

Hi halls, I've added some content to the OP.

 

[negation]

A linear transformation, f, has the property that f(\alpha u+ \beta v)= \alpha fu+ \beta fv where \alpha and \beta are scalars and u and v are vectors.

You know f(0,1) and f(1, 0) so given any (x, y), apply f to x(1, 0)+ y(0, 1).

Can you write (x,y) \in \mathbb{R}^2 as a linear combination of vectors for which you are given the value of f?

Suppose u = (e1) and v = (e2)
f(u) = (1,2,3) and f(v) = 0,-1,2)

f(λ₁u + λ₂v) = λ₁f(u) + λ₂f(v)
= λ₁ f(1,0) + λ₂ f(0,1)
∴λ₁(1,2,3) + λ₂ (0,-1,2) = λ₁(1,2,3) + λ₂(0,-1,2)

and if I am given λ₁ and λ₂ I can find a vector as a linear combination of
λ₁(1,2,3) + λ₂(0,-1,2).