\(B_1 = \left\{ \begin{bmatrix}2 \\ 1 \end{bmatrix}, \begin{bmatrix}3 \\ 2 \end{bmatrix} \right\} and B_2 = \left\{ \begin{bmatrix}3 \\ 1 \end{bmatrix}, \begin{bmatrix}7 \\ 2 \end{bmatrix} \right\}\) Consider \([B_2 : B_1] = \begin{bmatrix}3 & 7 &2 & 3 \\ 1 & 2 & 1 & 2 \end{bmatrix}\)
\(\sim \begin{bmatrix}1 & 2 & 1 & 2 \\ 3 & 7 & 2 & 3 \end{bmatrix} (R_1 \leftrightarrow R_2)\)
\(\sim \begin{bmatrix}1 & 2 & 1 & 2 \\ 0 & 1 & -1 & -3 \end{bmatrix} (R_1 \rightarrow R_2 - 3 R_1)\)
\(\sim \begin{bmatrix}1 & 0 & 3 & 8 \\ 0 & 1 & -1 & -3 \end{bmatrix} (R_1 \leftrightarrow R_1 - 2 R_2)\)
\(\displaystyle\therefore\) change of basis matrix from
\(\displaystyle{B}_{{1}}\to{B}_{{2}}{i}{s}\)
\(P = \begin{bmatrix} 3 & 8 \\ -1 & -3 \end{bmatrix}\)
\(men [v]_{B_1} = \begin{bmatrix} 2\\-1 \end{bmatrix}\)
\(men [v]_{B_2} = P[v]_{B_1} = \begin{bmatrix}3 & 8 \\-1 & -3 \end{bmatrix} \begin{bmatrix} 2\\-1 \end{bmatrix}\)
\(\therefore [v]_{B_2} = \begin{bmatrix} -2\\ 1\end{bmatrix}\)
Given the full and correct answer the two bases of B1
Given the full and correct answer the two bases of B1
Answers (1)
Relevant Questions
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 \(b= \begin{bmatrix} 5\\ 3 \end{bmatrix}\) are both (2,1) in the coordinate system defined by these two bases
(10%) In \(R^2\), there are two sets of coordinate systems, represented by two distinct bases: \((x_1, y_1)\) and \((x_2, y_2)\). If the equations of the same ellipse represented by the two distinct bases are described as follows, respectively: \(2(x_1)^2 -4(x_1)(y_1) + 5(y_1)^2 - 36 = 0\) and \((x_2)^2 + 6(y_2)^2 - 36 = 0.\) Find the transformation matrix between these two coordinate systems: \((x_1, y_1)\) and \((x_2, y_2)\).
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).
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}
(a) Find the bases and dimension for the subspace \(H = \left\{ \begin{bmatrix} 3a + 6b -c\\ 6a - 2b - 2c \\ -9a + 5b + 3c \\ -3a + b + c \end{bmatrix} ; a, b, c \in R \right\}\) (b) Let be bases for a vector space V,and suppose (i) Find the change of coordinate matrix from B toD. (ii) Find \(\displaystyle{\left[{x}\right]}_{{D}}{f}{\quad\text{or}\quad}{x}={3}{b}_{{1}}-{2}{b}_{{2}}+{b}_{{3}}\)
Solve the given Alternate Coordinate Systems and give a correct answer 10) Convert the equation from Cartesian to polar coordinates solving for \(r^2\):
\(\frac{x^2}{9} - \frac{y^2}{16} = 25\)
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 \(R^2, S = \left\{ \begin{bmatrix}1 \\ 0 \end{bmatrix}\begin{bmatrix} 0 \\1 \end{bmatrix} \right\}, v = \begin{bmatrix} 3 \\-2 \end{bmatrix} \)
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. \(\displaystyle{\left({b}\right)}{\left[{S}\right]}_{{{E},{C}}}\)
\(\displaystyle{\left({c}\right)}{\left[{S}\right]}_{{{B},{C}}}\)
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 \({x}_B = \begin{bmatrix}1 \\ -2 \end{bmatrix}\) find the vector x. e. If \(x = \begin{bmatrix} 2 \\ -4 \end{bmatrix} \ find\ {x}_B\)
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 \(R^2, S = \left\{ \begin{bmatrix}1 \\ 0 \end{bmatrix}\begin{bmatrix} 0 \\1 \end{bmatrix} \right\}, v = \begin{bmatrix} 3 \\-2 \end{bmatrix}\)
