What is the Result of Applying L to the Polynomial α+βτ+γτ2?

[sana2476]

Homework Statement

Let L : R3[τ] → R2[τ] be a linear transformation, where the bases for the polynomial vector spaces R3[τ] and R2[τ] are (1,τ,τ²) and (1,τ) respectively. We also know the matrix representation for L is:

A=[2 0 1]
[0 1 3]

What is the result of L(α+βτ+γτ²)?

The Attempt at a Solution

is it safe to say that identity matrix forms a basis? I need help understanding this problem

 

[dx]

What are the components of α+βτ+γτ² in the basis (1,τ,τ²)?

 

[sana2476]

The components would just be α=1, β=1, γ=1. Isn't that right?

 

[dx]

No.

Let's try a more familiar example. What are the components of the vector ai + bj + ck in the basis (i, j, k)?

 

[sana2476]

a,b,c would be the components.

 

[dx]

Yes, that's correct. Now what are the components of α1+βτ+γτ² in the basis (1, τ, τ²)?

 

[sana2476]

α,β,γ are the components in the basis (1, τ, τ²).

 

[dx]

Yes. Now, what do you get when the matrix

[2 0 1]
[0 1 3]

acts on the vector (α, β, γ)?

 

[sana2476]

you would get:

[2α + γ]
[β + 3γ]

Correct?

 

[dx]

Yes, you get the vector (2α + γ, β + 3γ). But what basis is this vector in?

 

[sana2476]

It's in the basis: (1, τ, τ²)?

 

[dx]

No. Read the question again. When you have a linear transformation L : A → B, and you want to represent L by a matrix, you must chose a basis for both A and B. What is the basis of B in this case?

 

[sana2476]

The basis for B is (1,τ)

 

[dx]

Ok, so what's the answer? What's L(α+βτ+γτ²)?

 

[sana2476]

Ok so (1,τ,τ2) is the basis for L(α+βτ+γτ²)

 

[dx]

I get the feeling you don't completely understand what a basis is. What is a basis?