# (a) Find the bases and dimension for the subspace

(a) Find the bases and dimension for the subspace $H=\left\{\left[\begin{array}{c}3a+6b-c\\ 6a-2b-2c\\ -9a+5b+3c\\ -3a+b+c\end{array}\right];a,b,c\in R\right\}$ (b) Let be bases for a vector space V,and suppose (i) Find the change of coordinate matrix from B toD. (ii) Find ${\left[x\right]}_{D}f\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}x=3{b}_{1}-2{b}_{2}+{b}_{3}$

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a)
$\left[\begin{array}{ccc}3& 6& -1\\ 6& -2& -2\\ -9& 5& 3\\ -3& 1& 1\end{array}\right]\sim \left[\begin{array}{ccc}3& 6& -1\\ 0& -14& 0\\ 0& 23& 0\\ 0& 7& 0\end{array}\right]by\left\{\begin{array}{l}{R}_{2}-2{R}_{1}\\ {R}_{3}+3{R}_{1}\\ {R}_{4}+{R}_{1}\end{array}$
$\left[\begin{array}{ccc}3& 6& -1\\ 0& 1& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]by\left\{\begin{array}{l}-\frac{1}{14}{R}_{2}\\ {R}_{3}-23{R}_{2}\\ {R}_{4}-7{R}_{2}\end{array}$ This shows that the vectors

${v}_{1}=\left[\begin{array}{c}3\\ 6\\ -9\\ -3\end{array}\right],{v}_{2}=\left[\begin{array}{c}6\\ -2\\ 5\\ 1\end{array}\right]$ are linearly independent and thus form bases for the given subspace. Simiarly ${v}_{2}=\left[\begin{array}{c}6\\ -2\\ 5\\ 1\end{array}\right],{v}_{3}=\left[\begin{array}{c}-1\\ 2\\ 3\\ 1\end{array}\right]$ also form basis for the set H, dimesion of the space is

2. b) Find the change of coordinate matrix from B to D. $\underset{D←B}{\underset{⏟}{P}}=\left[\begin{array}{ccc}2& 0& -3\\ -1& 3& 0\\ 1& 1& 2\end{array}\right]$

Find ${\left[x\right]}_{D}f\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}x=3{b}_{1}-2{b}_{2}+{b}_{3}.$
$\left[x{\right]}_{D}=\left[\begin{array}{ccc}2& 0& -3\\ -1& 3& 0\\ 1& 1& 2\end{array}\right]\left[\begin{array}{c}3\\ -2\\ 1\end{array}\right]=\left[\begin{array}{c}3\\ -9\\ 3\end{array}\right]$