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Can someone please check if my solution is correct? Thanks.

Considering the following basis: $$ B= \left\{ x^2,1+x^2,x-1 \right\}\\ C= \left\{ x,1-x,x^2 \right\} $$ a) Find the change of basis matrices from $B$ to $C$ and $C$ to $B$.

b) Find the coordinates of the vector $x+1$ in $C$.

c) Find a basis $D$ of $R^2[x]$ such that the change of basis matrix from $D$ to $C$ is the same as the change from $C$ to $B$.

My Solution:

a)To reach what I've done, I decomposed the vectors in one basis in terms of a linear combination of the spam of the other basis...

From $B$ to $C$: $$ \begin{pmatrix} 0 & 1 & 0\\ 0 & 1 & -1\\ 1 & 1 & 0\\ \end{pmatrix} $$ From $C$ to $B$: $$ \begin{pmatrix} -1 & 0 & 1\\ 1 & 0 & 0\\ 1 & -1 & 0\\ \end{pmatrix} $$

b) x+1 in $C$: $$ \alpha x + \beta (1-x) + \gamma x^2 = x+1\\ \gamma = 0\\ \alpha - \beta = 1\\ \beta = 1\\ \left[\begin{matrix} 2\\ 1\\ 0 \end{matrix}\right] $$

c)Basis $D$:

First vector is going to be the first column of this $C$ to $B$ matrix, times the vectors in basis $C$: $$ \left[\begin{matrix} -1\\ 1\\ 1\\ \end{matrix}\right] \cdot \left[\ x,1-x,x^2\right]\ = -x+1-x+x^2 = \\ x^2-2x+1 $$ Same for second and third vectors: $$ \left[\begin{matrix} 0\\ 0\\ -1\\ \end{matrix}\right] \cdot \left[\ x,1-x,x^2\right]\ = -x^2\\ \left[\begin{matrix} 1\\ 0\\ 0\\ \end{matrix}\right] \cdot \left[\ x,1-x,x^2\right]\ = x\\ $$ So the basis $D$ is going to be: $$ D= \left\{ x^2-2x+1,-x^2,x \right\} $$

Is everything correct?!

Thanks.

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Everything looks correct. The only thing I would say is that you should make sure D is in fact a basis by checking the vectors are linearly independent and checking whether they in fact span $R^2[x]$. This should be fairly easy by looking at the dimension of the subspace or the linear independence of the coordinates.

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