\begin{problem}{MATH 294}{SPRING 1987}{PRELIM 3}{9}{} \prob{For problems (a) - (c) use the bases $B$ and $B^{\prime}$ below:} \probnn{$B = \left\{ \left( \begin{array}{c} 1 \\ 0 \end{array} \right), \left( \begin{array}{c} 1 \\ -1 \end{array} \right) \right\}$ and $B^{\prime} = \left\{ \left( \begin{array}{c} 1 \\ 1 \end{array} \right), \left( \begin{array}{c} 0 \\ 1 \end{array} \right) \right\}$.} \probpart{Given that $[\vec{v}]_B=\left( \begin{array}{c} 2 \\ 3 \end{array} \right)$ what is $[\vec{v}]_{B^{\prime}}$?} \probpart{Using the standard relation between $\Re^2$ and points on the plane make a sketch with the point $\vec{v}$ clearly marked. Also mark the point $\vec{w}$, where $[\vec{w}]_B = \left( \begin{array}{c} 0 \\ -1 \end{array} \right)$. } \probpart{Draw the line defined by the points $\vec{v}$ and $\vec{w}$. Do the points on this line represent a subspace of $\Re^2$?} \end{problem} %---------------------------------- \begin{problem}{MATH 294}{SPRING 1987}{FINAL}{9}{} \prob{A general vector $\vec{v}$ in $\Re^2$ is $\vec{v} = b_1 \vec{v}_1 + b_2 \vec{v}_2 = b_1^{\prime} \vec{v}_1^{\, \prime} + b_2^{\prime} \vec{v}_2^{\,\prime}$. where} \probnn{$\vec{v}_1 = \left( \begin{array}{c} 1 \\ 0 \end{array} \right), \vec{v}_2 = \left( \begin{array}{c} 1 \\ 1 \end{array} \right), \vec{v}_1^{\, \prime} = \left( \begin{array}{c} 1 \\ 0 \end{array} \right), \vec{v}_2^{\, \prime} = \left( \begin{array}{c} 0 \\ 1 \end{array} \right)$.} \probnn{Find a matrix $_{B^{\prime}}[I]_{B}$ so that $\left( \begin{array}{c} b_1^{\prime} \\ b_2^{\prime} \end{array} \right) = _{B^{\prime}}[I]_{B} \left( \begin{array}{c} b_1 \\ b_2 \end{array} \right)$ for all vectors $\vec{v}$ in $\Re^2$.} \end{problem} %---------------------------------- \begin{problem}{MATH 293}{SPRING 1993}{FINAL}{5}{} \probb{Determine the matrix $H_{E,E}$ which represents reflection of vectors in $\Re^2$ about the y-axis in the standard basis $E = \left\{ \left( \begin{array}{c} 1 \\ 0 \end{array} \right), \left( \begin{array}{c} 0 \\ 1 \end{array} \right) \right\}$. Verify your answer by evaluating the expression $$ H_{E,E} \left( \begin{array}{c} x \\ y \end{array} \right). $$ } \probpart{Now consider a basis $B$ which is obtained by rotating each vector of the standard basis by 90 degrees in a counterclockwise direction. Find the change-of-basis matrices $(B:E)$ and $(E:B)$.} \probpart{Find $H_{B,B}$ from the formula $H_{B,B} = (E:B)H_{E,E}(B:E)$.} \probpart{It is claimed that $H_{B,B}$ is equal to the matrix $H_{E,E}$ which represents a reflection about the x-axis in the standard basis. Do you agree? Give geometrical reasons for your answer by drawing a suitable picture.} \end{problem} %---------------------------------- \begin{problem}{MATH 293}{FALL 1995}{PRELIM 3}{2}{} \prob{Consider the vector space $\Re^3$ and the three bases:} \probpartnn{the standard basis $E = \left\{ \left( \begin{array}{c} 1 \\ 0 \\ 0 \end{array} \right), \left( \begin{array}{c} 0 \\ 1 \\ 0 \end{array} \right), \left( \begin{array}{c} 0 \\ 0 \\ 1 \end{array} \right) \right\}$,} \probpartnn{the basis $B = \left\{ \left( \begin{array}{c} 1 \\ 0 \\ -1 \end{array} \right), \left( \begin{array}{c} 0 \\ 1 \\ -1 \end{array} \right), \left( \begin{array}{c} 1 \\ 2 \\ 3 \end{array} \right) \right\}$, and} \probpartnn{the basis $C = \left\{ \left( \begin{array}{c} 1 \\ 0 \\ 0 \end{array} \right), \left( \begin{array}{c} 1 \\ 1 \\ 0 \end{array} \right), \left( \begin{array}{c} 1 \\ 1 \\ 1 \end{array} \right) \right\}$.} \probpart{Given the $E$ coordinates of a vector $\vec{x}, [\vec{x}]_E = \left( \begin{array}{c} 1 \\ 2 \\ 3 \end{array} \right),$ find $[x]_C$.} \probpart{Given the $B$ coordinates of a vector $\vec{y}, [\vec{y}]_B = \left( \begin{array}{c} 1 \\ 0 \\ -1 \end{array} \right),$ find the coefficients $y_j$ in $\vec{y} = \vec{y}_1 \vec{e}_1 + \vec{y}_2 \vec{e}_2 + \vec{y}_3 \vec{e}_3$.} \probpart{Find the change-of-coordinates matrix $_C P _B$ whose columns consist of the $C$ coordinate vectors of the basis vectors of $B$.} \end{problem} %---------------------------------- \begin{problem}{MATH 293}{SPRING 1996}{PRELIM 3}{8}{} \prob{Let $$ B = \left\{ \left[ \begin{array}{c} -1 \\ 8 \end{array} \right] , \left[ \begin{array}{c} 1 \\ -5 \end{array} \right] \right\} , C = \left\{ \left[ \begin{array}{c} 1 \\ 4 \end{array} \right] , \left[ \begin{array}{c} 1 \\ 1 \end{array} \right] \right\}. $$} \probpart{Find the change of coordinate matrix from $B$ to $C$.} \probpart{Find the change of coordinate matrix from $C$ to $B$.} \end{problem} %---------------------------------- \begin{problem}{MATH 293}{SPRING 1996}{FINAL}{8}{} \prob{Let $$ B = \left\{ \left[ \begin{array}{c} 1 \\ 1 \end{array} \right] , \left[ \begin{array}{c} 2 \\ 0 \end{array} \right] \right\} , C = \left\{ \left[ \begin{array}{c} 2 \\ 2 \end{array} \right] , \left[ \begin{array}{c} 2 \\ -2 \end{array} \right] \right\}. $$} \probnn{Then the change of coordinates matrix from coordinates with respect to the basis $C$ to coordinates with respect to the basis $B$ is} \probpart{$\left( \begin{array}{cc} 2 & -2 \\ 0 & 2 \end{array} \right)$} \probpart{$\left( \begin{array}{cc} -4 & 4 \\ 0 & -4 \end{array} \right)$} \probpart{$\left( \begin{array}{cc} 0 & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{array} \right)$} \probpart{$\left( \begin{array}{cc} 0 & 4 \\ 4 & 4 \end{array} \right)$} \probpart{none of the above} \end{problem} %---------------------------------- \begin{problem}{MATH 294}{FALL 1995}{PRELIM 3}{8}{} \prob{You are given a vector space $V$ with an inner product $<,>$ and an orthogonal basis $B = \{ \vec{b}_1 , \vec{b}_2 , \vec{b}_3 , \vec{b}_4 , \vec{b}_5$ for $V$ for which $\left| \left| \vec{b}_i \right| \right| = 2, i = 1, \ldots, 5$. Suppose that $\vec{v}$ is in $V$ and $$ \left< \vec{v},\vec{b}_1\right> = \left< \vec{v},\vec{b}_2\right> = 0$$ $$ \left< \vec{v},\vec{b}_4\right> = 3, \left< \vec{v},\vec{b}_4\right> = 4, \left< \vec{v},\vec{b}_5\right> = 5$$ Find the coordinates of $\vec{v}$ with respect to the basis $B$ i.e. find $c_1,c_2,c_3,c_4,c_5$ such that $$ \vec{v} = c_1 \vec{b}_1 + c_2 \vec{b}_2 + c_3 \vec{b}_3 + c_4 \vec{b}_4 + c_5 \vec{b}_5$$} \end{problem} %---------------------------------- \begin{problem}{MATH 294}{SPRING 1998}{PRELIM 3}{3}{} \prob{Let $T:\wp_1 \rightarrow \wp_3$ be defined by \[ T\left[ p(t)\right] =t^2 p(t)\]and take \[ B = \left\{ 1,1+t \right\} \] to be the basis of $\wp_3$.} \probpart{Find the matrix of $T$ relative to the bases $B$ and $C$.} \probpart{Use this matrix to find $T[2+t]$.} \probpart{Let $E = \left\{1,t\right\}$ be the standard basis for $\wp_1$. Let $\left[ \vec{x} \right]_B$ be the coordinate vector of $\vec{x}$ in $\wp_1$ relative to the basis $B$, and let $\left[ \vec{x} \right]_E$ be the coordinate vector of $\vec{x}$ relative to the basis $E.$ What is the change of coordinate matrix $P.$ such that \[ P\left[ \vec{x} \right] _B = \left[ \vec{x} \right]_E. \] } \probnn{[Note: The result of part c) does not depend on the results of parts a) or b)] } \end{problem} %---------------------------------- \begin{problem}{MATH 294}{SPRING 1998}{Final}{4}{} \prob{In $P^2$, Find the change-of-coordinate matrix from the basis $$B = \left\{ 1-2t + t^2 , 3-5t,2t+3t^2\right\}$$ to the standard basis \[E = \{1,t,t^2\}. \] Then write $t^2$ as a linear combination of the polynomials in $B$, i.e. give the coordinates of $t^2$ with respect to the basis $B$. } \end{problem} %---------------------------------- \begin{problem}{MATH 294}{Fall 1998}{PRELIM 2}{3}{} \prob{Besides the standard basis $\varepsilon$ here are two bases for $\Re^2$: \[ B = \left\{ \underbrace { \left[ \begin{array}{c} 1 \\ 1 \\ \end{array} \right], }_ { b_1 } \underbrace { \left[ \begin{array}{c} -1 \\ 1 \\ \end{array} \right] }_ { b_2 } \right\} , C = \left\{ \underbrace { \left[ \begin{array}{c} 2 \\ 4 \\ \end{array} \right], }_{c_1} \underbrace { \left[ \begin{array}{c} -4 \\ 4 \\ \end{array} \right] }_{c_2} \right\} \] } \probpart{What vectors $\vec{x}$ are represented by $ \left[ \vec{x} \right]_B = \left[ \begin{array}{c} 2 \\ 14 \\ \end{array} \right] $ and $ \left[ \vec{x} \right]_C = \left[ \begin{array}{c} 2 \\ 4 \\ \end{array} \right] $? } \probpart{ Find a single tidy formula to find the components $ \left[ \begin{array}{c} d \\ e \\ \end{array} \right] $ of a vector $\vec{x}$ in the basis $B$ if you are given the components $ \left[ \begin{array}{c} f \\ g \\ \end{array} \right] $ of $\vec{x}$ in the basis $C$. } \probpart{ A student claims that the desired formula is $ \left[ \begin{array}{c} d \\ e \\ \end{array} \right] = \left[ \begin{array}{cc} 5 & -2 \\ 5 & 1 \\ \end{array} \right] \left[ \begin{array}{c} f \\ g \\ \end{array} \right]. $ Does this formula make the right prediction for the component vector $ \left[\vec{x}\right]_C = \left[ \begin{array}{c} f \\ g \\ \end{array} \right] = \left[ \begin{array}{c} 2 \\ 4 \\ \end{array} \right] $? } \end{problem} %---------------------------------- \begin{problem}{MATH 294}{FALL 1998}{Final}{6}{} \prob{Let $A = \left[ \begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array} \right].$} \probpart{Find orthogonal eigenvectors $\{ \vec{v}_1, \vec{v}_2 \}$ of $A$. [Hint: do not go on to parts d-e below until you have double checked that you have found two orthogonal unit vectors that are eigenvectors of $A$.]} \probpart{Use the eigenvectors above to diagonalize $A$.} \probpart{Make a clear sketch that shows the standard basis vectors $\{\vec{e}_1,\vec{e}_2\}$ of $\Re^2$ and the eigenvectors $\{\vec{v}_1,\vec{v}_2\}$ of $A$} \probpart{Give a geometric interpretation of the change of coordinates matrix, $P$, that maps coordinates of a vector with respect to the eigen basis to coordinates with respect to the standard basis.} \probpart{Let $\vec{b} = \left[ \begin{array}{c} 3 \\ 5 \end{array} \right]. $ Using orthogonal projection express $\vec{b}$ in terms of $\{\vec{v}_1,\vec{v}_2\}$ the eigenvectors of $A$.} \end{problem} %----------------------------------