# Consider the following vectors in R^4:v_

Question
Alternate coordinate systems

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

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

### Relevant Questions

Consider the following vectors in $$\displaystyle{R}^{{4}}:$$ $$v_1 = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix}, v_2 = \begin{bmatrix} 0 \\ 1 \\ 1 \\ 1 \end{bmatrix} v_3 = \begin{bmatrix}0 \\ 0 \\ 1 \\ 1 \end{bmatrix}, v_4 = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}$$ d. If $$x = \begin{bmatrix} 23 \\ 12 \\ 10 \\ 19 \end{bmatrix}, find \left\{ x\right\}_B e$$. If $${x}_B = \begin{bmatrix} 3 \\ 1 \\ -4 \\ -4 \end{bmatrix}$$, find x.

Consider the following two bases for $$\displaystyle{R}^{{3}}$$ :
$$\alpha := \left\{ \begin{bmatrix} 2 \\ 1\\ 3 \end{bmatrix}, \begin{bmatrix} -1 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix}3 \\ 1 \\ -1 \end{bmatrix} \right\} and\ \beta := \left\{ \begin{bmatrix}1 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix}-2 \\ 3\\ 1 \end{bmatrix}, \begin{bmatrix}2 \\ 3\\ -1 \end{bmatrix} \right\}$$ If $$[x]_{\alpha} = \begin{bmatrix}1 \\ 2 \\-1 \end{bmatrix}_{\alpha} then\ find\ [x]_{\beta}$$
(that is, express x in the $$\displaystyle\beta$$ coordinates).

Consider the linear transformation $$\displaystyle{U}:{R}^{{3}}\rightarrow{R}^{{3}}$$ defined by $$U \left(\begin{array}{c}x\\ y \\z \end{array}\right) = \left(\begin{array}{c} z - y \\ z + y \\ 3z - x - y \end{array}\right)$$ and the bases $$\epsilon = \left\{ \left(\begin{array}{c}1\\ 0 \\0\end{array}\right), \left(\begin{array}{c}0\\ 1 \\ 0\end{array}\right), \left(\begin{array}{c}0\\ 0 \\ 1\end{array}\right) \right\}, \gamma = \left\{ \left(\begin{array}{c}1 - i\\ 1 + i \\ 1 \end{array}\right), \left(\begin{array}{c} -1\\ 1 \\ 0\end{array}\right), \left(\begin{array}{c}0\\ 0 \\ 1\end{array}\right) \right\}$$, Compute the four coordinate matrices $$\displaystyle{{\left[{U}\right]}_{{\epsilon}}^{{\gamma}}},{{\left[{U}\right]}_{{\gamma}}^{{\gamma}}},$$

Consider the following linear transformation $$T : P_2 \rightarrow P_3$$, given by $$T(f) = 3x^2 f$$'. That is, take the first derivative and then multiply by $$3x^2$$ (a) Find the matrix for T with respect to the standard bases of $$P_n$$: that is, find $$[T]_{\epsilon}^{\epsilon}$$, where- $$\epsilon = {1, x, x^2 , x^n}$$ (b) Find N(T) and R(T). You can either work with polynomials or with their coordinate vectors with respect to the standard basis. Write the answers as spans of polynomials. (c) Find the the matrix for T with respect to the alternate bases: $$[T]_A^B$$ where $$A = {x - 1, x, x^2 + 1}, B = {x^3, x, x^2, 1}.$$

Let B and C be the following ordered bases of $$\displaystyle{R}^{{3}}:$$
$$B = (\begin{bmatrix}1 \\ 4 \\ -\frac{4}{3} \end{bmatrix},\begin{bmatrix}0 \\ 1 \\ 8 \end{bmatrix},\begin{bmatrix}1 \\ 1 \\ -2 \end{bmatrix})$$
$$C = (\begin{bmatrix}1 \\ 1 \\ -2 \end{bmatrix}, \begin{bmatrix}1 \\ 4 \\ -\frac{4}{3} \end{bmatrix}, \begin{bmatrix}0 \\ 1 \\ 8 \end{bmatrix})$$ Find the change of coordinate matrix I_{CB}

Consider the bases $$B = \left(\begin{array}{c}\begin{bmatrix}2 \\ 3 \end{bmatrix}, \begin{bmatrix}3 \\ 5 \end{bmatrix}\end{array}\right) of R^2 \ and\ C = \left(\begin{array}{c}\begin{bmatrix}1 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix}1\\0 \\ 1 \end{bmatrix}\end{array}, \begin{bmatrix}0 \\ 1\\1 \end{bmatrix}\right) of R^3$$.
and the linear maps $$S \in L (R^2, R^3) \ and\ T \in L(R^3, R^2)$$ given given (with respect to the standard bases) by $$[S]_{E, E} = \begin{bmatrix}2 & -1 \\ 5 & -3\\ -3 & 2 \end{bmatrix} \ and\ [T]_{E, E} = \begin{bmatrix}1 & -1 & 1 \\ 1 & 1 & -1 \end{bmatrix}$$ Find each of the following coordinate representations. $$\displaystyle{\left({b}\right)}{\left[{S}\right]}_{{{E},{C}}}$$
$$\displaystyle{\left({c}\right)}{\left[{S}\right]}_{{{B},{C}}}$$

Consider the bases $$B=(\begin{bmatrix}2 \\3 \end{bmatrix},\begin{bmatrix}3\\5 \end{bmatrix}) \ of \ r^{2} \ and \ E \ (\begin{bmatrix}1 \\1\\0 \end{bmatrix}, \begin{bmatrix}1 \\0\\1 \end{bmatrix}, \begin{bmatrix}0 \\0\\1 \end{bmatrix}) of \ R^{3}$$.
and the linear maps $$S\in \alpha (R^{2},R^{3}) \ and \ T \in\alpha (R^{3},R^{2})$$ given given (with respect to the standard bases) by $$[S]_{\epsilon,\epsilon}\begin{bmatrix}2 & -1 \\5 & -3\\-3 & 2 \end{bmatrix} \ and \ [T]_{\epsilon,\epsilon}\begin{bmatrix}1 & -1 & 1 \\1 & 1 & -1\end{bmatrix}.$$ Find each of the following coordinate representations. $$a) [S]_{\beta,\epsilon}, b) [S]_{\epsilon, E}, c) [S]_{\beta, E}$$

Consider the elliptical-cylindrical coordinate system (eta, psi, z), defined by $$x = a \ \cos h \ \eta \cos \psi, y = a \sin h\ \eta \sin \psi; z = z,\ \eta \ GE \ 0, 0 \ LE \ \psi LE \ 2 \pi, \ z R. In \ PS6$$
it was shown that this is an orthogonal coordinate system with scale factors $$\displaystyle{h}_{{1}}={h}_{{2}}={a}{\left({{\text{cosh}}^{{2}}\ }\eta-{{\cos}^{{2}}\psi}\right)}^{{{\frac{{{1}}}{{{2}}}}}}.$$
Determine the dual bases $$\displaystyle{\left({E}{1},{E}{2},{E}{3}\right)},{\left(\eta,\eta\psi,{z}\right)}.{S}{h}{o}{w}{t}\hat{:}{f}={a}\frac{{1}}{{a}}\frac{{\left({{\text{cosh}}^{{2}}{e}}{a}{t}-{{\cos}^{{s}}\psi}\right)}^{{1}}}{{2}}{\left[\frac{{f}}{\eta}{e}{1}+\frac{{f}}{\psi}{e}{2}+\frac{{f}}{{z}}{e}{3},\frac{{f}}{{w}}{h}{e}{r}{e}{\left({e}{1},{e}{2},{e}{3}\right)}\right.}$$ denotes the unit coordinate basis.

Interraption: To show that the system $$\displaystyle\dot{{r}}={r}{\left({1}-{r}^{2}\right)},\dot{\theta}={1}$$ is equivalent to $$\displaystyle\dot{{x}}={x}-{y}-{x}{\left({x}^{2}+{y}^{2}\right)},\dot{{y}}={x}+{y}-{y}{\left({x}^{2}+{y}^{2}\right)}$$ for polar to Cartesian coordinates.
The quadratic function $$\displaystyle{y}={a}{x}^{2}+{b}{x}+{c}$$ whose graph passes through the points (1, 4), (2, 1) and (3, 4).