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• Find All Values of $a$ which Will Guarantee that $A$ Has Eigenvalues 0, 3, and -3. Let $A$ be the matrix given by $A= \begin{bmatrix} -2 & 0 & 1 \\ -5 & 3 & a \\ 4 & -2 & -1 \end{bmatrix}$ for some variable $a$. Find all values of $a$ which will guarantee that $A$ has eigenvalues $0$, $3$, and $-3$.   Solution. Let $p(t)$ be the […]
• The set of $2\times 2$ Symmetric Matrices is a Subspace Let $V$ be the vector space over $\R$ of all real $2\times 2$ matrices. Let $W$ be the subset of $V$ consisting of all symmetric matrices. (a) Prove that $W$ is a subspace of $V$. (b) Find a basis of $W$. (c) Determine the dimension of $W$.   Proof. […]
• Given the Data of Eigenvalues, Determine if the Matrix is Invertible In each of the following cases, can we conclude that $A$ is invertible? If so, find an expression for $A^{-1}$ as a linear combination of positive powers of $A$. If $A$ is not invertible, explain why not. (a) The matrix $A$ is a $3 \times 3$ matrix with eigenvalues $\lambda=i , […] • Compute$A^5\mathbf{u}$Using Linear Combination Let $A=\begin{bmatrix} -4 & -6 & -12 \\ -2 &-1 &-4 \\ 2 & 3 & 6 \end{bmatrix}, \quad \mathbf{u}=\begin{bmatrix} 6 \\ 5 \\ -3 \end{bmatrix}, \quad \mathbf{v}=\begin{bmatrix} -2 \\ 0 \\ 1 \end{bmatrix}, \quad \text{ and } […] • If Two Vectors Satisfy A\mathbf{x}=0 then Find Another Solution Suppose that the vectors \[\mathbf{v}_1=\begin{bmatrix} -2 \\ 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \qquad \mathbf{v}_2=\begin{bmatrix} -4 \\ 0 \\ -3 \\ -2 \\ 1 \end{bmatrix}$ are a basis vectors for the null space of a$4\times 5$[…] • Linear Transformation$T:\R^2 \to \R^2$Given in Figure Let$T:\R^2\to \R^2$be a linear transformation such that it maps the vectors$\mathbf{v}_1, \mathbf{v}_2$as indicated in the figure below. Find the matrix representation$A$of the linear transformation$T$. Solution 1. From the figure, we see […] • Finitely Generated Torsion Module Over an Integral Domain Has a Nonzero Annihilator (a) Let$R$be an integral domain and let$M$be a finitely generated torsion$R$-module. Prove that the module$M$has a nonzero annihilator. In other words, show that there is a nonzero element$r\in R$such that$rm=0$for all$m\in M$. Here$r$does not depend on […] • A Relation of Nonzero Row Vectors and Column Vectors Let$A$be an$n\times n$matrix. Suppose that$\mathbf{y}$is a nonzero row vector such that $\mathbf{y}A=\mathbf{y}.$ (Here a row vector means a$1\times n$matrix.) Prove that there is a nonzero column vector$\mathbf{x}\$ such that $A\mathbf{x}=\mathbf{x}.$ (Here a […]