# Linear Algebra Transformations

1. May 26, 2015

### henry3369

1. The problem statement, all variables and given/known data
T(a0 + a1t+a2t2) = 3a0 + (5a0 - 2a1)t + (4a1 + a2)t2
Find the image of the vectors :
1. 1
2. t
3. t2

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

3. 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.

Last edited: May 26, 2015
2. May 26, 2015

### HallsofIvy

Staff Emeritus
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.