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• 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 […]
• Subset of Vectors Perpendicular to Two Vectors is a Subspace Let $\mathbf{a}$ and $\mathbf{b}$ be fixed vectors in $\R^3$, and let $W$ be the subset of $\R^3$ defined by \[W=\{\mathbf{x}\in \R^3 \mid \mathbf{a}^{\trans} \mathbf{x}=0 \text{ and } \mathbf{b}^{\trans} \mathbf{x}=0\}.\] Prove that the subset $W$ is a subspace of […]
• Determine whether the Given 3 by 3 Matrices are Nonsingular Determine whether the following matrices are nonsingular or not. (a) $A=\begin{bmatrix} 1 & 0 & 1 \\ 2 &1 &2 \\ 1 & 0 & -1 \end{bmatrix}$. (b) $B=\begin{bmatrix} 2 & 1 & 2 \\ 1 &0 &1 \\ 4 & 1 & 4 \end{bmatrix}$.   Solution. Recall that […]
• Find the Inverse Matrices if Matrices are Invertible by Elementary Row Operations For each of the following $3\times 3$ matrices $A$, determine whether $A$ is invertible and find the inverse $A^{-1}$ if exists by computing the augmented matrix $[A|I]$, where $I$ is the $3\times 3$ identity matrix. (a) $A=\begin{bmatrix} 1 & 3 & -2 \\ 2 &3 &0 \\ […]
• Eigenvalues and Algebraic/Geometric Multiplicities of Matrix $A+cI$ Let $A$ be an $n \times n$ matrix and let $c$ be a complex number. (a) For each eigenvalue $\lambda$ of $A$, prove that $\lambda+c$ is an eigenvalue of the matrix $A+cI$, where $I$ is the identity matrix. What can you say about the eigenvectors corresponding to […]
• An Orthogonal Transformation from $\R^n$ to $\R^n$ is an Isomorphism Let $\R^n$ be an inner product space with inner product $\langle \mathbf{x}, \mathbf{y}\rangle=\mathbf{x}^{\trans}\mathbf{y}$ for $\mathbf{x}, \mathbf{y}\in \R^n$. A linear transformation $T:\R^n \to \R^n$ is called orthogonal transformation if for all $\mathbf{x}, \mathbf{y}\in […]
• If the Augmented Matrix is Row-Equivalent to the Identity Matrix, is the System Consistent? Consider the following system of linear equations: \begin{align*} ax_1+bx_2 &=c\\ dx_1+ex_2 &=f\\ gx_1+hx_2 &=i. \end{align*} (a) Write down the augmented matrix. (b) Suppose that the augmented matrix is row equivalent to the identity matrix. Is the system consistent? […]
• Pullback Group of Two Group Homomorphisms into a Group Let $G_1, G_1$, and $H$ be groups. Let $f_1: G_1 \to H$ and $f_2: G_2 \to H$ be group homomorphisms. Define the subset $M$ of $G_1 \times G_2$ to be \[M=\{(a_1, a_2) \in G_1\times G_2 \mid f_1(a_1)=f_2(a_2)\}.\] Prove that $M$ is a subgroup of $G_1 \times G_2$.   […]