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I have a set

$S = \{ x^2 + 1, x + 1, 1 - x, x^3 \}$

in a polynomial vector space.

How do I write a vector matrix of $S$ in the base $B = \{ 1, x, x^2, x^3 \}$?

I attempted this using the formula: $M(S) = B^{-1} * T * B$, where $T$ is the matrix of the set.

I got the following result:

$$ T=\left(\begin{matrix} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1 \end{matrix}\right) $$ Is this correct or have I missed something?

I don't have the solutions so I cannot confirm it.

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1 Answer

up vote 2 down vote accepted

The purpose of this question is to write the elements of $S$ as linear combinations of elements of the basis $B$, then to give a matrix representation to the result.

Firstly let us call $(1,x,x^2,x^3) = (b_0,b_1,b_2,b_3).$ Then we have $$ x^2+1 = 1(1) + 0(x) + 1(x^2) + 0(x^3) = b_0+b_2\\\ x+1 = 1(1) + 1(x) + 0(x^2)+0(x^3) = b_0+b_1\\\ 1-x = 1(1) -1(x) + 0(x^2)+0(x^3) = b_0-b_1\\\ x^3 = 0(1) + 0(x) + 0(x^2) + 1(x^3) =b_3 $$ so that each element of $S$ is expressed as a linear combination of the basis elements in $B$. Then let us write each element of $S$ as a column vector in the basis $B$. We have $$ x^2+1 = b_0+b_2 = \left[\begin{array}{c}1\\0\\1\\0 \end{array} \right] $$ $$ x+1 = b_0+b_1 = \left[\begin{array}{c}1\\1\\0\\0\end{array} \right] $$ $$ 1-x = b_0-b_1 = \left[\begin{array}{c}1\\-1\\0\\0 \end{array} \right] $$ $$ x^3 = b_3 = \left[\begin{array}{c}0\\0\\0\\1 \end{array} \right] $$ and we then put all of these side by side in one matrix $$ M=\left[\begin{array}{cccc}1&1&1&0\\0&1&-1&0\\1&0&0&0\\0&0&0&1 \end{array}\right] $$ where $M$ is a matrix representation of $S$ in the basis $B$.

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