Linear Algebra Transformations

[henry3369]

Homework Statement

T(a₀ + a₁t+a₂t²) = 3a₀ + (5a₀ - 2a₁)t + (4a₁ + a₂)t²
Find the image of the vectors :
1. 1
2. t
3. t²

Homework Equations

T(a₀ + a₁t+a₂t²) = 3a₀ + (5a₀ - 2a₁)t + (4a₁ + a₂)t²

The Attempt at a Solution

I don't know how my book solves these transformations, but the answers are:
T(1) = 3+5t
T(t) = -2t+4t²
T(t²) = t²

How do you substitute a single vector for an entire expression to solve for each of these?
When it was a simple transformation (T(x) = x^2), you just replace x with the input, but for this one, you have to substitute an entire expression to find the transformation.

 

[HallsofIvy]

Homework Statement

T(a₀ + a₁t+a₂t²) = 3a₀ + (5a₀ - 2a₁)t + (4a₁ + a₂)t²
Find the image of the vectors :
1. 1
2. t
3. t²

This is just a matter of understanding and applying the given definition (and arithmetic).
You are told that T(a₀ + a₁t+a₂t²) = 3a₀ + (5a₀ - 2a₁)t + (4a₁ + a₂)t²
and asked to find T(1). 1= 1+ 0x+ 0x². a₀= 1, a₁= 0 and a₂= 0
T(1+ 0x+ 0x²)= 3(1)+ (5(1)- 2(0))t+ (4(0)+ 0)t²= 3+ 5t.

Similarly, t= 0+ 1t+ 0t² so a₀= 0, a₁= 1, and a₂= 0.

t²= 0+ 0t+ 1t² so a₀= 0, a₁= 0, and a₂= 1.

2. Homework Equations

T(a₀ + a₁t+a₂t²) = 3a₀ + (5a₀ - 2a₁)t + (4a₁ + a₂)t²

The Attempt at a Solution

I don't know how my book solves these transformations, but the answers are:
T(1) = 3+5t
T(t) = -2t+4t²
T(t²) = t²

How do you substitute a single vector for an entire expression to solve for each of these?
When it was a simple transformation (T(x) = x^2), you just replace x with the input, but for this one, you have to substitute an entire expression to find the transformation.