 # Since we will be using various bases Yulia 2020-12-30 Answered

Since we will be using various bases and the coordinate systems they define, let's review how we translate between coordinate systems. a. Suppose that we have a basis$B=\left\{{v}_{1},{v}_{2},\dots ,{v}_{m}\right\}f\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}{R}^{m}$. Explain what we mean by the representation {x}g of a vector x in the coordinate system defined by B. b. If we are given the representation ${\left\{x\right\}}_{B},$ how can we recover the vector x? c. If we are given the vector x, how can we find ${\left\{x\right\}}_{B}$? d. Suppose that BE is a basis for R^2. If ${x}_{B}=\left[\begin{array}{c}1\\ -2\end{array}\right]$ find the vector x. e. If

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a) Given is the coordinate vector of x Corresponding to the basis b) Let ${\left\{x\right\}}_{B}=\left[\begin{array}{c}{c}_{1}\\ {c}_{2}\\ {c}_{m}\end{array}\right]$ then
$x={c}_{1}{v}_{1}+{c}_{2}{v}_{2}+\dots +{c}_{m}{v}_{m},{c}_{i}\in R$ c) Of x is given and B be a basis for ${R}^{m}$ than x can be written as the linear combinations of vectors of B that is d) $B=\left\{=\left[\begin{array}{c}1\\ 3\end{array}\right],{\left\{x\right\}}_{B}=\left[\begin{array}{c}1\\ 1\end{array}\right]\right\}$ is a basis for
${R}^{2}$ ${\left\{x\right\}}_{B}=\left[\begin{array}{c}1\\ -2\end{array}\right]$
$x=1\left[\begin{array}{c}1\\ 3\end{array}\right]-2\left[\begin{array}{c}1\\ 1\end{array}\right]=\left[\begin{array}{c}-1\\ 1\end{array}\right]$
$\because x=\left[\begin{array}{c}-1\\ 1\end{array}\right]$ e) $x=\left[\begin{array}{c}2\\ -4\end{array}\right]=4\left[\begin{array}{c}1\\ 3\end{array}\right]+{c}_{2}\left[\begin{array}{c}1\\ 1\end{array}\right]=\left[\begin{array}{c}{c}_{1}+{c}_{2}\\ 3{c}_{1}+{c}_{2}\end{array}\right]$
$\because {c}_{1}+{c}_{2}=2,3{c}_{1}+{c}_{2}=-4$
$\because {c}_{1}=-3,{c}_{2}=5$
$\because {\left\{x\right\}}_{B}=\left[\begin{array}{c}-3\\ 5\end{array}\right]$