Algebra Help: Finding Change-of-Basis Matrices and Vectors

[JaysFan31]

Homework Statement

I'm working on change-of-basis matrices and vector spaces.

I have B=(1+x+x^2+x^3, 1+x+x^2, 1+x, 1) and C=(1+x^2, 1+x+2x^2, 1+4x+5x^2+x^3, -2+2x-x^2+5x^3).

Also, p(x)=3-8x+2x^2-6x^3.

I need to find [p]subscriptB and [p]subscriptC.

I need to show that the change of basis matrix from C to B times [p]subB=[p]subC and it doesn't work. I know the change of basis matrices.

Homework Equations

Nothing relevant that I can think of. It's probably what I'm missing. What exactly is the equation/formula for [p]subscriptB and [p]subscriptC?

The Attempt at a Solution

Here's my work for finding [p]subscript B:
I have the equation p(x)=3-8x+2x^2-6x^3 and B=(1+x+x^2+x^3, 1+x+x^2, 1+x, 1).
I said a(1+x+x^2+x^3)+b(1+x+x^2)+c(1+x)+d=3-8x+2x^2-6x^3.
I solved for a,b,c,d and got:
a=-6
b+a=2
a+b+c=-8
a+b+c+d=3

Thus, a=-6, b=8, c=-10, d=11.

So I said that [p]subscriptB is the vector:
[-6
8
-10
11]
, but this is obviously not right.

Any help or guidance would be greatly appreciated.

 

[mighty2000]

Dear student,

Thank you for your post. You are on the right track with your approach to finding [p]subscriptB. However, there are a few errors in your calculations.

Firstly, when you solved for a, b, c, and d, you made a mistake in the third equation. It should be a+b+c=-8, not a+b+c+d=-8. This mistake carries over to your calculated values for b, c, and d.

Secondly, the vector [p]subscriptB should be a column vector, not a row vector. This means that the values should be arranged vertically, not horizontally.

Lastly, when calculating [p]subscriptC, you need to use the change of basis matrix from B to C, not from C to B. This is because you are converting from the basis B to the basis C, not the other way around.

I hope this helps you in finding the correct values for [p]subscriptB and [p]subscriptC. Remember to pay attention to the order of operations and to use the correct change of basis matrix. Good luck with your work!