Give a full correct answer for given question 1- Let W be the set of all polynomials a + bt + ct^2 in P_{2} such that a + b + c = 0 Show that W is a subspace of P_{2} , find a basis for W, and then find dim(W) 2 - Find two different bases of R^{2} so that the coordinates of b= begin{bmatrix} 5 3 end{bmatrix} are both (2,1) in the coordinate system defined by these two bases

Question
Alternate coordinate systems
asked 2020-11-01
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

Answers (1)

2020-11-02
Consider \(\displaystyle{W}=\le{f}{t}{\left\lbrace{a}+{b}{t}+{c}{t}^{{{2}}},{a}+{b}+{c}={0}{r}{i}{g}{h}{t}\right\rbrace}\) Yes.W is a subspace of \(\displaystyle{P}_{{{2}}}.\) Since \(\displaystyle{0}\epsilon{W}.\) If \(\displaystyle{x}+{y}{t}+{z}{t}^{{{2}}}{\quad\text{and}\quad}{l}+{m}{t}+{n}{t}^{{{2}}}\) are elements of \(\displaystyle{P}_{{{2}}},\) then \(\displaystyle{\left({x}+{l}\right)}+{\left({y}+{m}\right)}{t}+{\left({z}+{n}\right)}{t}^{{2}}\epsilon{W}.\) So w is closed under vector addtion. Let \(\displaystyle{r}\epsilon{\mathbb{{{R}}}}\) and \(\displaystyle{a}+{b}{t}+{c}{t}^{{2}}\epsilon{W}\) then \(\displaystyle{r}{\left({a}+{b}{t}+{c}{t}{2}\right)}={r}{a}+{r}{b}{t}+{r}{c}{t}^{{2}}\epsilon{W}.\) Therefore w is closed under scalar multiplication . Therefore W is a subspace of \(\displaystyle{P}_{{2}}.\)
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Relevant Questions

asked 2020-12-30
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\(\displaystyle{B}={\left\lbrace{v}_{{1}},{v}_{{2}},\ldots,{v}_{{m}}\right\rbrace}{f}{\quad\text{or}\quad}{R}^{{m}}\). 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 \(\displaystyle{\left\lbrace{x}\right\rbrace}_{{B}},\) how can we recover the vector x? c. If we are given the vector x, how can we find \(\displaystyle{\left\lbrace{x}\right\rbrace}_{{B}}\)? d. Suppose that BE is a basis for R^2. If {x}_B = \begin{bmatrix}1 \\ -2 \end{bmatrix}ZSK find the vector x. e. If \(\displaystyle{x}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}\backslash-{4}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{f}\in{d}{\left\lbrace{x}\right\rbrace}_{{B}}\)
asked 2021-01-23
Give a correct answer for given question (A) Argue why { (1,0,3), (2,3,1), (0,0,1) } is a coordinate system ( bases ) for R^3 ? (B) Find the coordinates of (7, 6, 16) relative to the set in part (A)
asked 2021-02-21
All bases considered in these are assumed to be ordered bases. In Exercise, compute the coordinate vector of v with respect to the giving basis S for V. V is \(\displaystyle{R}^{{2}},{S}=\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}\backslash{0}{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}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace},{v}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{3}\backslash-{2}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}\\)
asked 2021-01-19
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
asked 2020-11-10
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}.
asked 2021-02-25
All bases considered in these are assumed to be ordered bases. In Exercise, compute coordinate vector v with respect to the giving basis S for V. V is \(\displaystyle{M}_{{22}},{S}=\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}&-{1}\backslash{0}&{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{0}&{1}\backslash{1}&{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}&{0}\backslash{0}&-{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}&{0}\backslash-{1}&{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace},{v}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}&{3}\backslash-{2}&{2}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}.\)
asked 2020-10-18
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)
asked 2021-02-25
Consider the bases \(\displaystyle{B}={\left({b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{c}\right\rbrace}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}\backslash{3}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{3}\backslash{5}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\right)}{o}{f}{R}^{{2}}{\quad\text{and}\quad}{C}={\left({b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{c}\right\rbrace}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}\backslash{1}\backslash{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}\backslash{0}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{0}\backslash{1}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}\right)}{o}{f}{R}^{{3}}\).
and the linear maps PSKS \in L (R^2, R^3) and T \in L(R^3, R^2) given given (with respect to the standard bases) by \(\displaystyle{\left[{S}\right]}_{{{E},{E}}}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}&-{1}\backslash{5}&-{3}\backslash-{3}&{2}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{\quad\text{and}\quad}{\left[{T}\right]}_{{{E},{E}}}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}&-{1}&{1}\backslash{1}&{1}&-{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}\) Find each of the following coordinate representations. \(\displaystyle{\left({a}\right)}{\left[{S}\right]}_{{{B},{E}}}\)
asked 2021-01-31
(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 \(\displaystyle{\left[{x}\right]}_{{D}}{f}{\quad\text{or}\quad}{x}={3}{b}_{{1}}-{2}{b}_{{2}}+{b}_{{3}}\)
asked 2021-02-14
All bases considered in these are assumed to be ordered bases. In Exercise, compute the coordinate vector of v with respect to the giving basis S for V. V is \(\displaystyle{M}_{{22}},{S}=\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}&{0}\backslash{0}&{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{0}&{0}\backslash{1}&{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{0}&{1}\backslash{0}&{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{0}&{0}\backslash{0}&{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace},{v}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}&{0}\backslash-{1}&{2}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}.\)
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