Question

Linear Algebra Question! (see attachment)

Answer 1

#5 (a) and (d)

Answer 2

For 5 (a), express 1 + x - x^3 as a linear combination of the basis vectors. Setup a coordinate vector using the coefficients of the linear combination in the order of the basis vectors.

Answer 3

Do you understand the isomorphism between an arbitrary vector space and F^n?

Answer 4

\[1 + x - x^3 = 1\cdot (1+ x) -1 \cdot (x^3) + 0\cdot (x^4+x^8)\]So we have the following coordinate vector:\[\left[ \begin {array}{c} 1\\ -1
\\ 0\end {array} \right] \]
Now transform the vector using the matrix:
\[\left[ \begin {array}{ccc} 2&9&-9\\ 0&14&-12
\\ 0&9&-7\end {array} \right]\left[ \begin {array}{c} 1\\ -1
\\ 0\end {array} \right] = \left[ \begin {array}{c} -7\\ -14
\\ -9\end {array} \right] \]
And take the coordinate vector and compute the vector using the basis:
\[-7\cdot (1 + x) -14 \cdot (x^3) - 9 \cdot (x^4 + x^8) = -7-7\,x-14\,{x}^{3}-9\,{x}^{4}-9\,{x}^{8}\]

Answer 5

For (d), assuming the matrix is diagonalizable, and it is, the basis will consist of the eigenvectors for the matrix.

Answer 6

We have:
\[ \left[ \begin {array}{c} 1\\ 4/3
\\ 1\end {array} \right] \,\, \left[ \begin {array}{c} 0\\ 1
\\ 1\end {array} \right]\,\, \left[ \begin {array}{c} 1\\ 0
\\ 0\end {array} \right] \]

Answer 7

good explanation, thanks, these make sense now