# Given the full and correct answer the two bases of B1

Question
Alternate coordinate systems

Given the full and correct answer the two bases of $$B1 = \left\{ \left(\begin{array}{c}2\\ 1\end{array}\right),\left(\begin{array}{c}3\\ 2\end{array}\right) \right\}$$
$$B_2 = \left\{ \left(\begin{array}{c}3\\ 1\end{array}\right),\left(\begin{array}{c}7\\ 2\end{array}\right) \right\}$$
find the change of basis matrix from $$\displaystyle{B}_{{1}}\to{B}_{{2}}$$ and next use this matrix to covert the coordinate vector
$$\overrightarrow{v}_{B_1} = \left(\begin{array}{c}2\\ -1\end{array}\right)$$ of v to its coodirnate vector
$$\overrightarrow{v}_{B_2}$$

2020-10-19

$$B_1 = \left\{ \begin{bmatrix}2 \\ 1 \end{bmatrix}, \begin{bmatrix}3 \\ 2 \end{bmatrix} \right\} and B_2 = \left\{ \begin{bmatrix}3 \\ 1 \end{bmatrix}, \begin{bmatrix}7 \\ 2 \end{bmatrix} \right\}$$ 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
$$\displaystyle{B}_{{1}}\to{B}_{{2}}{i}{s}$$
$$P = \begin{bmatrix} 3 & 8 \\ -1 & -3 \end{bmatrix}$$
$$men [v]_{B_1} = \begin{bmatrix} 2\\-1 \end{bmatrix}$$
$$men [v]_{B_2} = P[v]_{B_1} = \begin{bmatrix}3 & 8 \\-1 & -3 \end{bmatrix} \begin{bmatrix} 2\\-1 \end{bmatrix}$$
$$\therefore [v]_{B_2} = \begin{bmatrix} -2\\ 1\end{bmatrix}$$

### 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 $$R^2, S = \left\{ \begin{bmatrix}1 \\ 0 \end{bmatrix}\begin{bmatrix} 0 \\1 \end{bmatrix} \right\}, v = \begin{bmatrix} 3 \\-2 \end{bmatrix}$$