Consider the following vectors in R^4:v_

Consider the following vectors in R^4:v_

Question
Alternate coordinate systems
asked 2021-01-19

Consider the following vectors in \(\displaystyle{R}^{{4}}:\) \(v_1 = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix}, v_2 = \begin{bmatrix} 0 \\ 1 \\ 1 \\ 1 \end{bmatrix} v_3 = \begin{bmatrix}0 \\ 0 \\ 1 \\ 1 \end{bmatrix}, v_4 = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}\) a. Explain why \(B = \left\{ v_1, v_2, v_3, v_4 \right\}\)
forms a basis for \(\displaystyle{R}^{{4}}.\) b. Explain how to convert \(\left\{ x\right\}_B\) the representation of a vector x in the coordinates defined by B, into x, its representation in the standard coordinate system. c. Explain how to convert the vector x into,\(\displaystyle{\left\lbrace{x}\right\rbrace}_{{B}},\) its representation in the coordinate system defined by B

Answers (1)

2021-01-20

\(v_1 = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix}, v_2 = \begin{bmatrix} 0 \\ 1 \\ 1 \\ 1 \end{bmatrix} v_3 = \begin{bmatrix}0 \\ 0 \\ 1 \\ 1 \end{bmatrix}, v_4 = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}\) a) B form a basis if 1 B is L.T & 2 B spems \(\displaystyle{R}^{{4}}\) Check B is L.T \(\displaystyle\prec_{{1}}{\left({1},{1},{1},{1}\right)}+\prec_{{2}}{\left({0},{1},{1}{1}\right)}+\prec_{{3}}{\left({0},{0},{1},{1}\right)}+\prec_{{4}}{\left({0},{0},{0},{1}\right)}={0}\)
\(\displaystyle{\left(\prec_{{1}},+\prec_{{2}},\prec_{{1}}+\prec_{{2}}+\prec_{{3}},\prec_{{1}}+\prec_{{2}}+\prec_{{3}}+\prec_{{4}}\right)}={0}\)
\(\displaystyle\prec_{{1}}={0},\prec_{{2}}={0},\prec_{{3}}=\prec_{{4}}={0}\)
\(\Rightarrow v_1, v_2, v_3\ and\ v_4\ are\ L.T.\)

Also, span \(\displaystyle{R}^{{4}}.\)

b) Suppose \(\displaystyle{x}={\left({x}_{{1}},{x}_{{2}},{x}_{{3}},{x}_{{4}}\right)}{t}{h}{e}{n}{\left\lbrace{x}\right\rbrace}_{{B}}\)
\({x}_B = \begin{bmatrix} 1 & 0 & 0 & 0\\ 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} = \begin{bmatrix} x_1 \\ x_1 + x_2 \\ x_1 + x_2 + x_3 \\ x_1 + x_2 + x_3 + x_4 \end{bmatrix}\)

c)
\(\displaystyle{\left\lbrace{x}\right\rbrace}_{{B}}={x}_{{1}}{\left({1},{1},{1},{1}\right)}+{x}_{{2}}{\left({0},{1},{1},{1}\right)}+{x}_{{3}}{\left({0},{0},{1},{1}\right)}+{x}_{{4}}{\left({0},{0},{0},{1}\right)}\)
\(\displaystyle{\left\lbrace{x}\right\rbrace}_{{B}}={\left({x}_{{1}},{x}_{{1}}+{x}_{{2}},{x}_{{1}}+{x}_{{2}}+{x}_{{3}},{x}_{{1}}+{x}_{{2}}+{x}_{{3}}+{x}_{{4}}\right)}\)

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