\begin{problem}{MATH 294}{\spr{87}}{\pr{3}}{3}{} \prob{Consider the subspace of $C^2_\infty$ given by all things of the form $\vec{x}(t) = \vttwo{a\sin t + b\cos t}{c \sin t + d \cos t},$ where a,b,c \& d are arbitrary constants. Find a matrix representation of the linear transformation $T(\vec{x}) = D \vec{x}, \mbox{ where } D \vec{x} \equiv \dot{\vec{x}}.$ carefully define any terms you need in order to make this representation. Hint: A good basis for this vector space starts something like this $\left\{ \brc{\begin{array}{c} \sin t \\ 0 \end{array}} , \ldots \right\}.$} \end{problem} %---------------------------------- \begin{problem}{MATH 294}{\spr{87}}{\pr{3}}{5}{} \prob{The idea of eigenvalue $\lambda$ and eigenvector $\bv{v}$ can be generalized from matrices and $\Re^n$ to linear transformations and their related vector spaces. If $T(\bv{v}) = \lambda \bv{v}$ (and $\bv{v} \neq 0)$ then $\lambda$ is an eigenvalue of $T$, and $\bv{v}$ is its associated eigenvector.} \probnn{For the subspace of $\bv{x}(t)$ in $C^1_\infty$ with $\bv{x}(0) = \bv{x}(1) = 0$ find \emph{an} eigenvalue and eigenvector of $T(\bv{x}) = D^2 \bv{x},$ where $D^2 \bv{x} \equiv \ddot{\bv{x}} - \sqbrc{\artwo{2}{0}{0}{3}} \bv{x}$. What is the kernel of $T$?} \end{problem} %---------------------------------- \begin{problem}{MATH 294}{spr{97}}{FINAL}{2}{} \prob{$T$ is linear transformation from $C^2_\infty$ to $C^2_\infty$ which is given by $T(\bv{x}) = \dot{\bv{x}}$} \end{problem} %---------------------------------- \begin{problem}{MATH 294}{\fa{87}}{\pr{3}}{14}{} \prob{Find the kernel of the linear transformation $T(\bv{x}(t)) \equiv \vttwo{x_1^\prime}{x^\prime_2} - \sqbrc{\artwo{0}{1}{-4}{0}} \vttwo{x_1}{x_2}$ where $T$ transforms $C^2_\infty$ into $C^2_\infty$} \end{problem} %---------------------------------- \begin{problem}{MATH 294}{\fa{97}}{\pr{3}}{5}{} \prob{Define $T \brc{\vtthree{x}{y}{z}} \equiv \vtthree{x+y}{x-z}{y+z},$ which is a linear transformation of $\Re^3$ into itself.} \probpart{Is $T$ 1-1?} \probpart{Is $T$ onto?} \probpart{Is $T$ an isomorphism?} \probnn{Substantiate your answers.} \end{problem} %---------------------------------- \begin{problem}{MATH 294}{\fa{87}}{FINAL}{1}{} \prob{$T$ is a linear transformation of $\Re^3$ into $\Re^2$ such that $T \vtthree{1}{-1}{2} = \vttwo{2}{1} , \; T\vtthree{2}{1}{0} = \vttwo{1}{0}, \; T \vtthree{1}{1}{1} = \vttwo{1}{-1}.$} \probpart{Is $T$ 1-1?} \probpart{Determine the matrix of $T$ relative to the standard bases in $\Re^3$ and $\Re^2$.} \end{problem} %---------------------------------- \begin{problem}{MATH 294}{\fa{87}}{FINAL}{7a}{} \prob{Consider the boundary-value problem} \probnn{$X^{\prime \prime} + \lambda X = 0, \; \; 0 < x < \pi, \; X(0) = X(\pi) = 0,$ where $\lambda$ is a given real number.} \probpart{Is the set of all solutions of this problem a subspace of $C_\infty[0,\pi]?$ why?} \probpart{Let $W =$ set of all functions $X(x)$ in $C_infty[0,\pi]$ such that $X(0) = X(\pi) = 0.$ Is $T \equiv D^2 - \lambda$ linear as a transformation of $W$ into $C_\infty[0,\pi]?$ Why?} \probpart{For what values of $\lambda$ is $Ker(T)$ nontrivial?} \probpart{Choose one of those values of $\lambda$ and determine $Ker(T)$} \end{problem} %---------------------------------- \begin{problem}{MATH 294}{\fa{89}}{\pr{3}}{3}{} \prob{Let $W$ be the following subspace of $\Re^3,$ $W = Comb\brc{\vtthree{1}{0}{1}, \vtthree{1}{1}{-1}, \vtthree{2}{1}{0}, \vtthree{3}{3}{-3}}.$} \probpart{Show that $\vtthree{1}{0}{1}, \vtthree{1}{1}{-1}$ is a basis for $W$.} \probnn{For b) and c) below, let $T$ be the following linear transformation $T: W \to \Re^3,$, $T\brc{\vtthree{w_1}{w_2}{w_3}} = \sqbrc{\arthree{1}{0}{-1}{0}{0}{0}{0}{0}{0} \vtthree{w_1}{w_2}{w_3}}$ for those $w = \vtthree{w_1}{w_2}{w_3}$ in $\Re^3$ which belong to $W$.}\probnn{[You are allowed to use a) even if you did not solve it.]} \probpart{What is the dimension of $Range(T)$? (Complete reasoning, please.} \probpart{What is the dimension of $Ker(T)$? (Complete reasoning, please.} \end{problem} %---------------------------------- \begin{problem}{MATH 294}{\fa{89}}{FINAL}{7}{} \prob{Let $T: \Re^2 \to \Re^2$ be the linear transformation given in the standard basis for $\Re^2$ by $T\brc{\vttwo{x}{y}} = \vttwo{x+y}{0}.$} \probpart{Find the matrix of $T$ in the standard basis for $\Re^2$}\probpart{Show that $\beta = \brc{\vttwo{1}{1}, \vttwo{1}{2}}$ is also a basis for $\Re^2$.} \probnn{In c) below, you may use the result of b) even if you did not show it.} \probpart{Find the matrix of $T$ in the basis $\beta$ given in b). (I.e., in $T: \Re^2 \to \Re^2$ both copies of $\Re^2$ have the basis $\beta$.} \end{problem} %---------------------------------- \begin{problem}{MATH 294}{\spr{90?}}{\pr{2}}{4}{} \prob{Let $A$ be a linear transformation from a vector space $V$ to another vector space $U$. Let $\brc{\vec{v}_1, \ldots, \vec{v}_n}$ be a basis for $V$ and let $\brc{\vec{u}_1, \ldots, \vec{u}_n}$ be a basis for $U$.} \probnn{Suppose it is known that } \probpartnn{$A(\vec{v}_1) = 2 u_2$} \probpartnn{$A(\vec{v}_2) = 3 u_3$}\probpartnn{$\vdots$} \probpartnn{$A(\vec{v}_i) = (i+1) \vec{u}_{i+1}$}\probpartnn{$\vdots$} \probpartnn{$A(\vec{v}_{n-1}) = n \vec{u}_n$} \probnn{and $A(\vec{v}_n) = 0 \leftarrow$ zero vector in $U$.} \probnn{Can you find $A(\vec{v})$ in terms of the $\vec{u}_i$'s where $\vec{v} = \vec{v}_1 + \vec{v}_2 + \ldots + \vec{v}_n = \sumn \vec{v}_i$} \end{problem} %---------------------------------- \begin{problem}{MATH 294}{\fa{91}}{FINAL}{8}{} \prob{T/F} \partinc \partinc \probpart{If $T: V \to W$ is a linear transformation, then the range of $T$ is a subspace of $V$.} \probpart{If the range of $T: V \to W$ is $W$, then $T$ is 1-1. } \probpart{If the null space of $T: V \to W$ is $\{0\}$, then $T$ is 1-1.} \probpart{Every change of basis matrix is a product of elementary matrices.} \probpart{If $T: U \to V$ and $S: V \to W$ are linear transformations, and $S$ is not 1-1, then $ST : U \to W$ is not 1-1.} \probpart{If $V$ is a vector space with an inner product, (,), if $\{ \vec{w}_1, \vec{w}_2, \ldots, \vec{w}_n \}$ is an orthonormal basis for $V$, and if $\vec{v}$ is a vector in $V$, then $\vec{v} = \sumn(\vec{v},\vec{w}_i)\vec{w}_i.$} \probpart{$T: V_n \to V_n$ is an isomorphism if and only if the matrix which represents $T$ in any basis is non-singular.} \probpart{If $S$ and $T$ are linear transformations of $V_n$ into $V_n$, and in a given basis, $S$ is represented by a matrix $A$, and $T$ is represented by a matrix $B$, then $ST$ is represented by the matrix $AB$ } \probnn{-note- Matrices are not necessarily square.} \end{problem} %---------------------------------- \begin{problem}{MATH 294}{\spr{92}}{\pr{3}}{5}{} \prob{The vector space $V_3$ has the standard basis $S = \brc{\vec{e}_1, \vec{e}_2, \vec{e}_3}$ and the basis $B = \brc{2 \vec{e}_2, -\frac{1}{2} \vec{e}_1, \vec{e}_3 }.$} \probpart{Find the change of basis matrices $(B:S)$ and $(S:B).$ If a vector $\vec{v}$ has the representation $\vtthree{1}{1}{1}$ in the standard basis, find its representation $\beta(\vec{v})$ in the $B$ basis.} \probpart{A transformation $T$ is defined as follows: $T \vec{v} =$ the reflection of $\vec{v}$ across the $x-z$ plane in the standard basis. (For reflection, in $V_2$ the reflection of $a \vec{i} + b \vec{j}$ across the x axis would be $a \vec{i} - b \vec{j}.$ Find a formula for $T$ in the standard basis. Why is $T$ a linear transformation?} \probpart{Find $T_B$, the matrix of $T$ in the $B$ basis.} \probpart{Interpret $T$ geometrically in the $B$ basis, i.e., describe $T_B$ in terms of rotations, reflections, etc.} \end{problem} %---------------------------------- \begin{problem}{MATH 294}{\fa{92}}{FINAL}{6}{} \prob{Let $C^2(-\infty, \infty)$ be the vector space of twice continuously differentiable functions on $-\infty < x <\infty$ and $C^0(\-infty, \infty)$ be the vector space of continuous functions on $-\infty < x <\infty$.} \probpart{Show that the transformation $L: C^2(\infty,\infty) \to C^0(-\infty,\infty)$ defined by $L y = \ndpd{y}{x} - 4 y$ is linear.} \probpart{Find a basis for the null space of $L$. Note: You must show that the vectors you choose are linearly independent. } \end{problem} %---------------------------------- \begin{problem}{MATH 293}{\spr{95}}{FINAL}{2}{} \prob{Let $P^3$ be the vector space of polynomials of degree $\leq 3$, and let $L: P^3 \to P^3$ be given by $L(p)(t) = t \ndpd{p}{t}(t) + 2 p(t).$} \probpart{Show that $L$ is a linear transformation.} \probpart{Find the matrix of $L$ in the basis $\brc{1, t, t^2, t^3}.$} \probpart{Find a solution of the differential equation $t \ndpd{p}{t} + 2 p(t) = t^3.$ Do you think that you have found the general solution?} \end{problem} %---------------------------------- \begin{problem}{MATH 293}{\spr{95}}{FINAL}{3}{} \prob{Let $V$ be the vector space of real $3 \times 3$ matrices.} \probpart{Find a basis of $V$. What is the dimension of $V$?} \probnn{Now consider the transformation $L: V \to V$ given by $L(A) = A + A^T$.} \probpart{Show that $L$ is a linear transformation.} \probpart{Find a basis for the null space (kernel) of $L$.} \end{problem} %---------------------------------- \begin{problem}{MATH 294}{\spr{97}}{FINAL}{10}{} \prob{Let $P_2$ be the vector space of polynomials of degree $\leq 2$, equipped with the inner product $= \int_{-1}^1 p(t) q(t) d t$ 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}