Question

Find the change of coordinates matrix from the basis B = (1, t, (3/2)t^2 - 1/2) to the standard basis in P2 (1, t, t^2) - can anyone help me please? I have a test tomorrow!

Answer 1

Take every element of basis B. Write its coordinates wrt basis P2. You will get 3 vectors. Use these as 3 column vectors to form a matrix

Answer 2

So I would have b1 = p1, b2 = p2, and b3 = 3/2p3 - 1/2?

Answer 3

How do I turn that into a matrix?

Answer 4

\[\left[\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ \frac{1}{3} & 0 & \frac{2}{3}\end{matrix}\right] * \left(\begin{matrix}1 \\ t \\ (3/2)t^2 - 1/2\end{matrix}\right)=\left(\begin{matrix}1 \\ t \\ t^2\end{matrix}\right)\]
Doesn't that imply that the matrix you seek is the one on the left?

Answer 5

See for the first element in B
ur coordinates wrt P2 are 1,0,0

Answer 6

And the inverse transformation would be the inverse of that matrix? It's invertible too :P

Answer 7

ohhh wow thanks malevolence

Answer 8

Sorry, I was thinking about it instead of writing it down, that's what took so long lol Do you see why that is the matrix you need?

Answer 9

Yes, I do...

Answer 10

Ok, then another question that is somewhat related

Answer 11

\[P=\begin{bmatrix}
1&0&1/3\\
0&1&0\\
0&0&2/3
\end{bmatrix}\\E=BP\]

Answer 12

Okay, I just left out the entire thought process haha
If you have a matrix that you want to change the rows/columns or add/subtract them or multiply by a constant you simply DO THAT TO THE IDENTITY MATRIX and then multiply it to the original, that's all I did the find the matrix.

Answer 13

given the original basis for B, what is the B-coordinate vector for 4-t+t^2?

Answer 14

i got this crazy answer of Xb = 13/3, -1, 2/3

Answer 15

Apply the inverse change of basis to that coordinate vector. Since you're going from E to B this time instead of vice versa.