Linear transformation

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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 Rm
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
 
Last edited:

Answers and Replies

  • #2
HallsofIvy
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A linear transformation, f, has the property that [itex]f(\alpha u+ \beta v)= \alpha fu+ \beta fv[/itex] where [itex]\alpha[/itex] and [itex]\beta[/itex] 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).
 
  • #3
pasmith
Homework Helper
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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 [itex](x,y) \in \mathbb{R}^2[/itex] as a linear combination of vectors for which you are given the value of [itex]f[/itex]?
 
  • #4
818
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A linear transformation, f, has the property that [itex]f(\alpha u+ \beta v)= \alpha fu+ \beta fv[/itex] where [itex]\alpha[/itex] and [itex]\beta[/itex] 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.
 
  • #5
818
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A linear transformation, f, has the property that [itex]f(\alpha u+ \beta v)= \alpha fu+ \beta fv[/itex] where [itex]\alpha[/itex] and [itex]\beta[/itex] 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 [itex](x,y) \in \mathbb{R}^2[/itex] as a linear combination of vectors for which you are given the value of [itex]f[/itex]?


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).
 

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