Since we will be using various bases

Question

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 $B = {v_{1}, v_{2}, \dots, v_{m}} f or R^{m}$ . 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 ${x}_{B},$ how can we recover the vector x? c. If we are given the vector x, how can we find ${x}_{B}$ ? d. Suppose that BE is a basis for R^2. If $x_{B} = [\begin{matrix} 1 \\ - 2 \end{matrix}]$ find the vector x. e. If $x = [\begin{matrix} 2 \\ - 4 \end{matrix}] f i n d x_{B}$

Answer 1

a) Given $B = {v_{1}, v_{2} . . . v_{m}} i s a b a s i s f o r R^{m} {x}_{B}$ is the coordinate vector of x Corresponding to the basis $B t h a t i s o f x = c_{1} v_{1} + c_{2} v_{2} + c_{3} v_{3} + . . . + c_{m} v_{m} t h a n {x}_{B} = [\begin{matrix} c_{1} \\ c_{2} \\ c_{m} \end{matrix}]$ b) Let ${x}_{B} = [\begin{matrix} c_{1} \\ c_{2} \\ c_{m} \end{matrix}]$ 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 $x = c_{1} v_{1} + c_{2} v_{2} + . . . + c_{m} v_{m}, c i \in R t h e n {x}_{B} = [\begin{matrix} c_{1} \\ c_{2} \\ c_{m} \end{matrix}]$ d) $B = {= [\begin{matrix} 1 \\ 3 \end{matrix}], {x}_{B} = [\begin{matrix} 1 \\ 1 \end{matrix}]}$ is a basis for
$R^{2}$ ${x}_{B} = [\begin{matrix} 1 \\ - 2 \end{matrix}]$
$x = 1 [\begin{matrix} 1 \\ 3 \end{matrix}] - 2 [\begin{matrix} 1 \\ 1 \end{matrix}] = [\begin{matrix} - 1 \\ 1 \end{matrix}]$
$∵ x = [\begin{matrix} - 1 \\ 1 \end{matrix}]$ e) $x = [\begin{matrix} 2 \\ - 4 \end{matrix}] = 4 [\begin{matrix} 1 \\ 3 \end{matrix}] + c_{2} [\begin{matrix} 1 \\ 1 \end{matrix}] = [\begin{matrix} c_{1} + c_{2} \\ 3 c_{1} + c_{2} \end{matrix}]$
$∵ c_{1} + c_{2} = 2, 3 c_{1} + c_{2} = - 4$
$∵ c_{1} = - 3, c_{2} = 5$
$∵ {x}_{B} = [\begin{matrix} - 3 \\ 5 \end{matrix}]$

Since we will be using various bases

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