= \int_{-1}^1 p(t) q(t) d t\] Let $T: P_2 \to P_2$ be the transformation which sends the polynomial $p(t)$ to the polynomial \[ (1-t^2) p^{\prime \prime}(t) - 2 t p ^{\prime}(t)+6p(t) \]} \probpart{Show that $T$ is linear.} \probpart{Verify that $T(1) = 6$ and $T(t) = 4 t.$ Find $T(t^2)$.}\probpart{Find the matrix $A$ of $T$ with respect to the standard basis $\epsilon = \brc{1,t,t^2}$ for $P_2.$} \probpart{Find the basis for $Nul(A)$ and $Col(A)$.} \probpart{Use the Gram-Schmidt process to find an orthogonal basis $B$ for $P_2$ starting form $\epsilon.$} \end{problem} %---------------------------------- \begin{problem}{MATH 294}{\fa{97}}{\pr{3}}{5}{} \prob{Let $T: \Re^2 \to \Re^2$ be the linear transformation that rotates every vector (starting at the origin) by $\theta$ degrees in the counterclockwise direction. Consider the following two bases for $\Re^2$: \[ B = \brc{\vttwo{1}{0}, \vttwo{0}{1}}, \] and \[ C = \brc{\vttwo{\cos \alpha}{\sin \alpha}, \vttwo{-\sin \alpha}{\cos \alpha}}. \]} \probpart{Find the matrix $[T]_B$ of $T$ in the standard basis $B$.}\probpart{Find the matrix $[T]_C$ of $T$ in the basis $C$. Does $[T]_C$ depend on the angle $\alpha$?} \end{problem} %---------------------------------- \begin{problem}{MATH 294}{\fa{97}}{FINAL}{9}{} \prob{Consider the vector space $V$ of $2 \time 2$ matrices. Define a transformation $T: V \to V$ by $T(A) = A^T,$ where $A$ is an element of $V$ (that is, it is a $2 \times 2$ matrix), and $A^T$ is the transpose of $A$.} \probpart{Show that $T$ is linear transformation.} \probnn{The value $\lambda$ is an \emph{eigenvalue} for $T$, and $\vec{v} \neq 0$ is the corresponding eigenvector, if $T(\vec{v}) = \lambda \vec{v}$. (\emph{Note}: here $\vec{v}$ is a $2 \times 2$ matrix).} \probpart{Find an eigenvalue of $T$ (You need only find one, not all of them). (\emph{Hint}: Search for matrices $A$ such that $T(A)$ is a scalar multiple of $A$.)} \probpart{Find an eigenvector for the particular eigenvalue that yl=ou found in part (b).} \probpart{Let $W$ be the complete eigenspace of $T$ with the eigenvalue from part (b) above. Find a basis for $W$. What is the dimension of $W$?} \end{problem} %---------------------------------- \begin{problem}{MATH 294}{\spr{98}}{FINAL}{6}{} \prob{Let $T: P^2 \to P^3$ be the transformation that maps the second order polynomial $p(t)$ into $(1+2 t) p (t),$} \probpart{Calculate $T(1), T(t),$ and $T(t^2)$.} \probpart{Show that $T$ is a linear transformation.} \probpart{Write the components of $T(1), T(t), T(t^2)$ with respect to the basis $C = \{1,t,t^2, 1+t^3\}.$} \probpart{Find the matrix of $T$ relative to the bases $B = \{1,t,t^2\}$ and $C = \{ 1, t, t^2, 1+t^3\}$.} \end{problem} %---------------------------------- \begin{problem}{MATH 294}{\fa{98}}{\pr{3}}{1}{} \prob{Consider the following three vectors in $\Re^3$ \[ \vec{y} = \vtthree{1}{0}{1}, \vec{u}_1 = \vtthree{1}{1}{1}, \mbox{and} \vec{u}_2 = \vtthree{1}{-1}{0}. \] [Note: $\vec{u}_1$ and $\vec{u}_2$ are orthogonal.].} \probpart{Find the orthogonal projection of $\vec{y}$ onto the subspace of $\Re^3$ spanned by $\vec{u}_1$ and $\vec{u}_2$.} \probpart{What is the distance between $\vec{y}$ and $span\{\vec{u}_1, \vec{u}_2\}$?} \probpart{In terms of the standard basis for $\Re^3,$ find the matrix of the linear transformation that orthogonally projects vectors onto $span\{\vec{u}_1, \vec{u}_2 \}.$} \end{problem} %---------------------------------- \begin{problem}{MATH 294}{\fa{98}}{FINAL}{4}{} \prob{Here we consider the vector spaces $P_1, P_2, $ and $P_3$ (the spaces of polynomials of degree 1,2 and 3).} \probpart{Which of the following transformations are linear? (Justify your answer.)} \probparts{$T: P_1 \to P_3, \; T(p) \equiv t^2 p(t) + p(0)$} \probparts{$T: P_1 \to P_1, \; T(p) \equiv p(t) + t$} \probpart{Consider the linear transformation $T: P_2 \to P_2$ defined by $T(a_0 + a_1 t + a_2 t^2) \equiv (-a_1 + a_2) + (-a_0 + a_1) t + (a_2) t^2. $ with respect to the standard basis of $P_2, \beta = \{ 1, t , t^2\}$, is $A = \sqbrc{\arthree{0}{-1}{1}{-1}{1}{0}{0}{0}{1}}$. Note that an eigenvalue/eigenvector pair of $A$ is $\lambda =1, v = \vtthree{0}{1}{1}.$ Find an eigenvaue/eigenvector (or eigenfunction) pair of $T$. That is, find $\lambda$ and $g(t)$ in $P_2$ such that $T(g(t)) = \lambda g(t).$} \probpart{Is the set of vectors in $P_2 \{ 3+t, -2+t, 1+t^2 \}$ a basis of $P_2$? (Justify your answer.)} \end{problem} %---------------------------------- \begin{problem}{MATH 293}{\spr{?}}{\pr{2}}{4}{} \prob{Let $M$ be the transformation from $P^n$ to $P^n$ such that \[ M p(t) = \frac{1}{2} [p(t) + p(-t)] \mbox{(t real)}\]} \probpart{If $n = 3$ find the matrix of this transformation with respect to the basis $\{ 1,t,t^2, t^3 \}.$} \probpart{Let $N = I-M.$ What is $N p(t)$ in terms of $p(t)$? Show that $M^2 = MM = M$, $MN = MN = 0$} \end{problem} %---------------------------------- \begin{problem}{MATH 294}{\fa{87}}{\pr{2}}{3 MAKE-UP}{} \probb{If $A$ is an $n \times n$ matrix with $rank(A) = r$, then what is the dimension of the vector space of all solutions of the system of linear equations $A \vec{x} = \vec{0}$ } \probpart{What is the dimension of the kernel of the linear transformation from $\Re^n$ to $\Re^n$ which has $A$ for its matrix in the standard basis. } \end{problem} %---------------------------------- \begin{problem}{MATH 294}{\fa{87}}{\pr{2}}{14 MAKE-UP}{} \prob{Show that if $T:V \to W$ is a linear transformation from $V$ to $W$, and kernel$(T) = \vec{0},$ then $T$ is 1-1. (Recall: kernel$(T) = \set{\vv \in V|\; T(\vv)=\vec{0}}.)$} \end{problem} %---------------------------------- \begin{problem}{MATH 294}{\fa{87}}{FINAL}{6 MAKE-UP}{} \prob{Let $T: \Re^2 \to \Re^4$ be a linear transformation.} \probpart{If $T \vttwo{2}{7}=\vtfour{3}{1}{0}{2} \and T\vttwo{3}{-1} = \vtfour{-1}{0}{1}{0},$ what is $T\vttwo{-9}{26}?$} \probpart{What are $T\vttwo{1}{0} \and T\vttwo{0}{1}?$} \probpart{What is the matrix of $T$ in the basis $\vttwo{1}{1}, \vttwo{1}{-1}$ for $\Re^2,$ and the standard basis for $\Re^4$?} \end{problem} %---------------------------------- \begin{problem}{MATH 294}{SUMMER 1989}{\pr{2}}{1}{} \probb{} \probpart{Find a basis for $ker(L)$, where $L$ is linear transformation from $\Re^4$ to $\Re^3$ defined by \[ L(\vec{x}) \equiv \sqbrc{\begin{array}{rrrr} 1 & 2 & -4 & 3 \\ 1 & 2 & -2 & 2 \\ 2 & 4 & -2 & 3 \end{array}} \vtfour{x_1}{x_2}{x_3}{x_4} \]} \probpart{What is the dimension of $ker(L)?$} \probpart{Is the vector $\vec{y} = \vtthree{1}{1}{2}$ in $range(L)?$ (Justify your answer.) If so, find all vectors $\vec{x}$ in $\Re^4$ which satisfy $L(\vec{x}) = \vec{y}$} \end{problem} %---------------------------------- \begin{problem}{MATH 294}{SUMMER 1989}{\pr{2}}{4}{} \prob{Let $P$ be the linear transformation from $\Re^3$ to $\Re^3$ defined by \[ P \vtthree{x}{y}{z} = \vtthree{x}{y}{0}. \]} \probpart{Find a basis for $ker(P).$} \probpart{Find a basis for $range(P).$}\probpart{Find all vectors $\vec{x}$ in $\Re^3$ such that $P \vec{x} = \vtthree{1}{2}{0}.$} \probpart{Find all vectors $\vec{x}$ in $\Re^3$ such that $P \vec{x} = \vtthree{1}{2}{3}.$} \end{problem} %---------------------------------- \begin{problem}{MATH 293}{SPRING 1995}{\pr{3}}{4}{} \prob{Let $L_\theta: \Re^2 \to \Re^2$ be the linear transformation which represent orthogonal projection onto the line $\ell_\theta$ forming angle $\theta$ with the x-axis.$$\centerline{\epsfxsize=2.4in \epsfbox{2_8_142.jpg}}$$} %------------------------------------------------------------------------------------------------------picture p. 142 \probpart{Find the matrix $T$ of $L_\theta$ (with respect to the standard basis of $\Re^2$).} \probpart{Is $L_\theta$ invertible. Explain your answer geometrically.} \probpart{Find all the eigenvalues of $T$.} \end{problem} %---------------------------------- \begin{problem}{MATH 294}{\fa{98}}{\pr{2}}{1}{} \prob{The unit square $OBCD$ below gets mapped to the parallelogram $OB^{\prime} C^{\prime} D^\prime$ (on the $x_1 - x_3$ plane) by the linear transformation $T: \Re^2 \to \Re^3$ shown.$$\centerline{\epsfxsize=2.4in \epsfbox{2_8_145.jpg}}$$} %------------------------------------------------------------------------------------------------------picture p. 145 \probnn{Problems (b) - (e) below can be answered with or without use of the matrix $A$ from part (a).} \probpart{Is this transformation one-to-one? For this and all other short answer questions on this test, some explanation is needed.)} \probpart{What is the null space of $A$?} \probpart{What is the column space of $A$?} \probpart{Is $A$ invertible? (No need to find the inverse if it exists.)} \end{problem} %---------------------------------- \begin{problem}{MATH 294}{FALL ?}{FINAL}{1 MAKE-UP}{} \prob{Consider the homogeneous system of equations $B \vec{x} = \vec{0}$, where \[ B = \sqbrc{\begin{array}{rrrrr} 0 & 1 & 0 & -3 & 1 \\ 2 & -1 & 0 & 3 & 0 \\ 2 & -3 & 0 & 0 & 4 \end{array}}, \; \vec{x} = \vtfive{x_1}{x_2}{x_3}{x_4}{x_5}, \and \vec{0} = \vtthree{0}{0}{0}.\]}\probpart{Find a basis for the subspace $W \subset \Re^5,$ where $W =$ set of all solutions of $B \vec{x} = \vec{0}. $} \probpart{Is $B$ 1-1 (as a transformation of $\Re^5 \to \Re^3$)? Why?} \probpart{Is $B: \Re^5 \to \Re^3$ onto? Why?} \probpart{Is the set of all solutions of $B \vec{x} = \vtthree{3}{0}{0}$ a subspace of $\Re^5$? Why?} \end{problem}