# Let beta = (x^{2} - x, x^{2} + 1,x - 1), beta' = (x^2 - 2x - 3, - 2x^2 + 5x + 5, 2x^2 - x - 3) be ordered bases for P_2(C). Find the change of coordinate matrix Q that changes beta ' -coordinates into beta -coordinates. Question
Alternate coordinate systems 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. 2021-02-09
LET A = $$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{1}&{1}&{0}&{1}&-{2}&{2}\backslash{h}{l}\in{e}-{1}&{0}&{1}&-{2}&{5}&-{1}\backslash{h}{l}\in{e}{0}&{1}&-{1}&-{3}&{5}&-{3}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$ It may be observed that the entries in the columns of A are the coefficients of x2, x and the scalar multiples of 1 in the ordered bases β and β’ respectively. To determine the change of coordinates matrix from $$\displaystyle\beta’\to\beta,$$ we will reduce A to its RREF as under: Add 1 times the 1st row to the 2nd row Add -1 times the 2nd row to the 3rd row Multiply the 3rd row by -1/2 Add -1 times the 3rd row to the 2nd row Add -1 times the 2nd row to the 1st row The RREF of A is $$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{1}&{0}&{0}&{3}&-{6}&{3}\backslash{h}{l}\in{e}{0}&{1}&{0}&-{2}&{4}&-{1}\backslash{h}{l}\in{e}{0}&{0}&{1}&{1}&-{1}&{2}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$ Then the change of coordinates matrix from \beta’ to \beta is Q= $$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{3}&-{6}&{3}\backslash{h}{l}\in{e}-{2}&{4}&-{1}\backslash{h}{l}\in{e}{1}&-{1}&{2}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$

### Relevant Questions 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 A and C be the following ordered bases of P3(t): $$\displaystyle{A}={\left({1},{1}+{t},{1}+{t}+{t}^{{2}},{1}+{t}+{t}^{{2}}+{t}^{{3}}\right)}$$
$$\displaystyle{C}={\left({1},-{t},{t}^{{2}}-{t}^{{3}}\right)}$$ Find tha change of coordinate matrix $$\displaystyle{I}_{{{C}{A}}}$$ Let B and C be the following ordered bases of $$\displaystyle{R}^{{3}}:$$
$$\displaystyle{B}={\left({b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}\backslash{4}\backslash-{\frac{{{4}}}{{{3}}}}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{0}\backslash{1}\backslash{8}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}\backslash{1}\backslash-{2}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}\right)}$$
$$\displaystyle{C}={\left({b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}\backslash{1}\backslash-{2}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}\backslash{4}\backslash-{\frac{{{4}}}{{{3}}}}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{0}\backslash{1}\backslash{8}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}\right)}$$ Find the change of coordinate matrix I_{CB} Give a full correct answer for given question 1- Let W be the set of all polynomials $$\displaystyle{a}+{b}{t}+{c}{t}^{{2}}\in{P}_{{{2}}}$$ such that $$\displaystyle{a}+{b}+{c}={0}$$ Show that W is a subspace of $$\displaystyle{P}_{{{2}}},$$ find a basis for W, and then find dim(W) 2 - Find two different bases of $$\displaystyle{R}^{{{2}}}$$ so that the coordinates of $$\displaystyle{b}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{5}\backslash{3}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}$$ are both (2,1) in the coordinate system defined by these two bases Let $$\displaystyle{B}=\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}\backslash-{2}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace}{\quad\text{and}\quad}{C}=\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}\backslash{3}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace}$$ be bases for R^2. Find the change-of-coordinate matrix from B to C. 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}. (a) Find the bases and dimension for the subspace $$\displaystyle{H}=\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{3}{a}+{6}{b}-{c}\backslash{6}{a}-{2}{b}-{2}{c}\backslash-{9}{a}+{5}{b}+{3}{c}\backslash-{3}{a}+{b}+{c}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{a},{b},{c}\in{R}{r}{i}{g}{h}{t}\right\rbrace}$$ (b) Let be bases for a vector space V,and suppose (i) Find the change of coordinate matrix from B toD. (ii) Find $$\displaystyle{\left[{x}\right]}_{{D}}{f}{\quad\text{or}\quad}{x}={3}{b}_{{1}}-{2}{b}_{{2}}+{b}_{{3}}$$ Let $$\displaystyle{B}=\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}\backslash-{2}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace}{\quad\text{and}\quad}{C}=\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}\backslash{3}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace}$$
be bases for$$\displaystyle{R}^{{2}}.$$ Change-of-coordinate matrix from C to B. Given the elow bases for R^2 and the point at the specified coordinate in the standard basis as below, (40 points) $$\displaystyle{B}{1}=\le{f}{t}{\left\lbrace{\left({1},{0}\right)},{\left({0},{1}\right)}{r}{i}{g}{h}{t}\right\rbrace}&{M}{S}{K}{B}{2}={\left({1},{2}\right)},{\left({2},-{1}\right)}{r}{i}{g}{h}{t}\rbrace{\left({1},{7}\right)}={3}^{\cdot}{\left({1},{2}\right)}-{\left({2},{1}\right)}{N}{S}{K}{B}{2}={\left({1},{1}\right)},{\left(-{1},{1}\right)}{\left({3},{7}={5}^{\cdot}{\left({1},{1}\right)}+{2}^{\cdot}{\left(-{1},{1}\right)}{N}{S}{K}{B}{2}={\left({1},{2}\right)},{\left({2},{1}\right)}{\left({0},{3}\right)}={2}^{\cdot}{\left({1},{2}\right)}-{2}^{\cdot}{\left({2},{1}\right)}{N}{S}{K}{\left({8},{10}\right)}={4}^{\cdot}{\left({1},{2}\right)}+{2}^{\cdot}{\left({2},{1}\right)}{N}{S}{K}{B}{2}={\left({1},{2}\right)},{\left(-{2},{1}\right)}{\left({0},{5}\right)}={N}{S}{K}{\left({1},{7}\right)}=\right.}$$ a. Use graph technique to find the coordinate in the second basis. (10 points) b. Show that each basis is orthogonal. (5 points) c. Determine if each basis is normal. (5 points) d. Find the transition matrix from the standard basis to the alternate basis. (15 points) Consider the following two bases for $$\displaystyle{R}^{{3}}$$ :
$$\displaystyle\alpha\:=\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}\backslash{1}\backslash{3}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}-{1}\backslash{0}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{3}\backslash{1}\backslash-{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace}{\quad\text{and}\quad}\beta\:=\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}\backslash{1}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}-{2}\backslash{3}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}\backslash{3}\backslash-{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace}$$ If $$\displaystyle{\left[{x}\right]}_{{\alpha}}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}\backslash{2}\backslash-{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}_{{\alpha}}{t}{h}{e}{n}{f}\in{d}{\left[{x}\right]}_{{\beta}}$$
(that is, express x in the $$\displaystyle\beta$$ coordinates).