# To explain^ Why the beta-coordinate vectors of beta = {b_{1}, ... , b_{n}} are the columns e_{1}, ... , e_{n} of the n times n identity matrix.

Question
Alternate coordinate systems
To explain^ Why the beta-coordinate vectors of $$\beta = {b_{1}, ... , b_{n}}$$
are the columns $$e_{1}, ... , e_{n}$$
of the $$n \times n$$ identity matrix.

2021-02-20
Consider a vector x in V such that, $$x= c_{1} b_{1} + c_{2} b_{2} + ... + c_{n} b_{n}$$
The coordinates of x relative to basis $$\beta = {b_{1}, b_{2}, ... , b_{n}}$$ ,
also called beta-coordinates of x are given by, $$[x]_{\beta}=\begin{bmatrix}c_{1} \\...\\c_{n} \end{bmatrix}$$
Since $$\beta = {b_{1}, ..., b_{n}}$$ from a basis for V.
Thus, the vectors $${b_{1}, ..., b_{n}} are linearly independent. Therefore, if any vector b_k is to be written in terms of \(b_{1}, ..., b_{n}$$
That is, an arbitrary vector $$b_{k}$$
can be written as $$b_{k} = 0 \cdot b_{1}, ... + 1 \cdot b_{k} + ... + 0 \cdot b_{n}.$$
Here, k varies from 1 to n.
Thus, the beta-coordinates of $$b_{1}, ..., b_{n}$$
in this case are $$\left\{\begin{bmatrix}c_{1} \\...\\c_{n} \end{bmatrix}\right\}$$
The matrix formed by these beta-coordinates is $$\begin{bmatrix}1 & \cdots & 0 \\ \cdots & \cdots & \cdots \\ 0 & \cdots & 1 \end{bmatrix}$$
That is, beta-coordinate vectors of $$\beta = {b_{1}, ..., b_{n}}$$
are the columns $$e_{1}, ..., e_{n}$$ of the
n \cdot n identity matrix.

### Relevant Questions

A subset $${u_{1}, ..., u_{p}}$$ in V is linearly independent if and only if the set of coordinate vectors
$${[u_{1}]_{\beta}, ..., [u_{p}]_{\beta}}$$
is linearly independent in $$R^{n}$$
Consider the following vectors in $$\displaystyle{R}^{{4}}:$$ $$\displaystyle{v}_{{1}}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}\backslash{1}\backslash{1}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{v}_{{2}}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{0}\backslash{1}\backslash{1}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{v}_{{3}}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{0}\backslash{0}\backslash{1}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{v}_{{4}}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{0}\backslash{0}\backslash{0}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}$$ a. Explain why $$\displaystyle{B}=\le{f}{t}{\left\lbrace{v}_{{1}},{v}_{{2}},{v}_{{3}},{v}_{{4}}{r}{i}{g}{h}{t}\right\rbrace}$$
forms a basis for $$\displaystyle{R}^{{4}}.$$ b. Explain how to convert $$\displaystyle\le{f}{t}{\left\lbrace{x}{r}{i}{g}{h}{t}\right\rbrace}_{{B}},$$ 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
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}.
The change - of - coordinate matrix from $$\mathscr{B} = \left\{\begin{bmatrix}3\\-1\\4\\\end{bmatrix}\begin{bmatrix}2\\0\\ -5 \\\end{bmatrix}\begin{bmatrix}8\\-2\\7\\ \end{bmatrix}\right\}$$
to the standard basis in $$RR^{n}.$$
The vector x is in $$H = Span \ {v_{1}, v_{2}}$$
and find the beta-coordinate vector $$[x]_{\beta}$$
Let $$\displaystyle\gamma={\left\lbrace{t}^{{2}}-{t}+{1},{t}+{1},{t}^{{2}}+{1}\right\rbrace}{\quad\text{and}\quad}\beta={\left\lbrace{t}^{{2}}+{t}+{4},{4}{t}^{{2}}-{3}{t}+{2},{2}{t}^{{2}}+{3}\right\rbrace}{b}{e}{\quad\text{or}\quad}{d}{e}{r}{e}{d}{b}{a}{s}{e}{s}{f}{\quad\text{or}\quad}{P}_{{2}}{\left({R}\right)}.$$ Find the change of coordinate matrix Q that changes $$\displaystyle\beta{c}\infty{r}{d}\in{a}{t}{e}{s}\int{o}\gamma-{c}\infty{r}{d}\in{a}{t}{e}{s}$$
Let $$\displaystyle\beta={\left({x}^{{{2}}}-{x},{x}^{{{2}}}+{1},{x}-{1}\right)},\beta'={\left({x}^{{2}}-{2}{x}-{3},-{2}{x}^{{2}}+{5}{x}+{5},{2}{x}^{{2}}-{x}-{3}\right)}$$ be ordered bases for $$\displaystyle{P}_{{2}}{\left({C}\right)}.$$ Find the change of coordinate matrix Q that changes $$\displaystyle\beta'$$ -coordinates into $$\displaystyle\beta$$ -coordinates.
$$A 0 = 0$$
$$A(cv) = cAv$$
$$A(v\ +\ w) = Av\ +\ Aw$$
Suppose A = QR is a QR factorization of an $$\displaystyle{m}\times{n}$$ matrix A (with linearly independent columns). Partition A as $$\displaystyle{\left[{A}_{{1}}{A}_{{2}}\right]},{w}{h}{e}{r}{e}{A}_{{1}}$$ has p columns. Show how to obtain a QR factorization of $$\displaystyle{A}_{{1}}$$, and explain why your factorization has the appropriate properties.