# Linear Algebra Question (Change of Basis)

"In $\mathbb{P}_2$, find the change-of-coordinate matrix from the basis $B=\{1-3t^2, 2+t-5t^2, 1+2t\}$ to the standard basis. Then write $t^2$ as a linear combination of the polynomials in $B$."

I believe the CoC matrix to be: $\left(\begin{array}{ccc}1 &2& 1\\ 0& 1& 2\\ -3 &-5& 0\end{array}\right)$? Can someone confirm this? I also do not understand the second part of the question.

I assume that $\mathbb{P}_2$ is the vector space of all polynomials of degree at most $2$ and real coefficients (you are probably not aware of this, but there is no absolute standard notation for this vector space; you should specify what vector space you mean, and what it is a vector space "over").

If $\mathbf{V}$ is any vector space, and $\beta=[\mathbf{v}_1,\ldots,\mathbf{v}_n]$ and $\gamma=[\mathbf{w}_1,\ldots,\mathbf{w}_n]$ are two bases for $\mathbf{V}$, how does one find the change-of-coordinates matrix from $\beta$ to $\gamma$?

1. We write each vector in $\gamma$ in terms of the vectors in $\beta$.
2. The expression for $\mathbf{w}_1$ gives the first column of the matrix; the expression for $\mathbf{w}_2$ gives the second column of the matrix, etc.

Caveat. There is some disagreement in nomenclature between books; sometimes, what I called the "change of coordinates matrix from $\beta$ to $\gamma$" is called the change of coordinate matrix from $\gamma$ to $\beta$. Check your book/notes! If that is the case, you would need to go the other way: write each vector in $\beta$ in terms of the vectors in $\gamma$.

So here, we would want to write each of the standard basis vectors $1$, $t$, and $t^2$, in terms of the vectors in $B$, $1-3t^2$, $2+t-5t^2$, and $1+2t$.

For instance, doing a quick linear system of equations, we have that $$1 = 10(1-3t^2) -6(2+t-5t^2) + 3(1+2t),$$

So, the change of coordinate matrix will have first column with $10$, $-6$, and $3$. So the change-of-coordinate matrix will look like: $$B = \left(\begin{array}{rrr} 10& \Box & \Box\\ -6 & \Box & \Box\\ 3 & \Box & \Box \end{array}\right).$$ Now fill in the rest of the boxes.

(Alternatively, find the change of coordinates matrix from the standard basis to $B$, which is very easy to find, and then find its inverse. That will be the matrix you want).

Caveat: If my warning above applies (that is, you use the other convention), then you instead need to write each of the vectors in $B$ in terms of the standard basis instead (this is easy to do, though).

To write $t^2$ as a linear combination of the polynomials in $B$, just use the fact that the coordinate vector of $t^2$ relative to the standard basis is $(0,0,1)^t$ (transpose) , and multiply by the matrix you found.

Caveat: If my warning above applies (that is, if you use the other convention), you would first need to find the inverse of the matrix you found in order to do this.

Unfortunately, given the problem I'm not really sure what convention you use. It would make more sense for the matrix you find in the first part to be useful directly to solve the second part; which suggests the convention I used above (which is not the one I normally use). On the other hand, if you use that convention, then the second part of the problem is something you need to do first anyway, so the second part would be superfluous. Please double-check what convention you use.

An alternative method for the second part of the question (supposing you couldn't find the change of basis matrix) is to solve the equation $c_1(1-3t^2)+c_2(2+t-5t^2)+c_3(1+2t)=t^2$. By comparing coefficients of $t^0$, $t^1$, and $t^2$, you can write three equations and solve. Note that this method takes a linear combination of elements from the basis $B$ and equates it to the vector you want to find. The values for the constants give you $Rep_B(t^2)$.