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Purdue University Linear Algebra Exam Problems and Solutions


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  • Annihilator of a Submodule is a 2-Sided Ideal of a RingAnnihilator of a Submodule is a 2-Sided Ideal of a Ring Let $R$ be a ring with $1$ and let $M$ be a left $R$-module. Let $S$ be a subset of $M$. The annihilator of $S$ in $R$ is the subset of the ring $R$ defined to be \[\Ann_R(S)=\{ r\in R\mid rx=0 \text{ for all } x\in S\}.\] (If $rx=0, r\in R, x\in S$, then we say $r$ annihilates […]
  • Are These Linear Transformations?Are These Linear Transformations? Define two functions $T:\R^{2}\to\R^{2}$ and $S:\R^{2}\to\R^{2}$ by \[ T\left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} 2x+y \\ 0 \end{bmatrix} ,\; S\left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = \begin{bmatrix} x+y […]
  • Powers of a Matrix Cannot be a Basis of the Vector Space of MatricesPowers of a Matrix Cannot be a Basis of the Vector Space of Matrices Let $n>1$ be a positive integer. Let $V=M_{n\times n}(\C)$ be the vector space over the complex numbers $\C$ consisting of all complex $n\times n$ matrices. The dimension of $V$ is $n^2$. Let $A \in V$ and consider the set \[S_A=\{I=A^0, A, A^2, \dots, A^{n^2-1}\}\] of $n^2$ […]
  • Find a basis for $\Span(S)$, where $S$ is a Set of Four VectorsFind a basis for $\Span(S)$, where $S$ is a Set of Four Vectors Find a basis for $\Span(S)$ where $S= \left\{ \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} , \begin{bmatrix} -1 \\ -2 \\ -1 \end{bmatrix} , \begin{bmatrix} 2 \\ 6 \\ -2 \end{bmatrix} , \begin{bmatrix} 1 \\ 1 \\ 3 \end{bmatrix} \right\}$.   Solution. We […]
  • Complex Conjugates of Eigenvalues of a Real Matrix are EigenvaluesComplex Conjugates of Eigenvalues of a Real Matrix are Eigenvalues Let $A$ be an $n\times n$ real matrix. Prove that if $\lambda$ is an eigenvalue of $A$, then its complex conjugate $\bar{\lambda}$ is also an eigenvalue of $A$.   We give two proofs. Proof 1. Let $\mathbf{x}$ be an eigenvector corresponding to the […]
  • The Rank of the Sum of Two MatricesThe Rank of the Sum of Two Matrices Let $A$ and $B$ be $m\times n$ matrices. Prove that \[\rk(A+B) \leq \rk(A)+\rk(B).\] Proof. Let \[A=[\mathbf{a}_1, \dots, \mathbf{a}_n] \text{ and } B=[\mathbf{b}_1, \dots, \mathbf{b}_n],\] where $\mathbf{a}_i$ and $\mathbf{b}_i$ are column vectors of $A$ and $B$, […]
  • Row Equivalent Matrix, Bases for the Null Space, Range, and Row Space of a MatrixRow Equivalent Matrix, Bases for the Null Space, Range, and Row Space of a Matrix Let \[A=\begin{bmatrix} 1 & 1 & 2 \\ 2 &2 &4 \\ 2 & 3 & 5 \end{bmatrix}.\] (a) Find a matrix $B$ in reduced row echelon form such that $B$ is row equivalent to the matrix $A$. (b) Find a basis for the null space of $A$. (c) Find a basis for the range of $A$ that […]
  • Quiz 2. The Vector Form For the General Solution / Transpose Matrices. Math 2568 Spring 2017.Quiz 2. The Vector Form For the General Solution / Transpose Matrices. Math 2568 Spring 2017. (a) The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution. \[ \left[\begin{array}{rrrrr|r} 1 & 0 & -1 & 0 &-2 & 0 \\ 0 & 1 & 2 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 \\ \end{array} \right].\] […]

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