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• Prove that $(A + B) \mathbf{v} = A\mathbf{v} + B\mathbf{v}$ Using the Matrix Components Let $A$ and $B$ be $n \times n$ matrices, and $\mathbf{v}$ an $n \times 1$ column vector. Use the matrix components to prove that $(A + B) \mathbf{v} = A\mathbf{v} + B\mathbf{v}$.   Solution. We will use the matrix components $A = (a_{i j})_{1 \leq i, j \leq n}$, $B = […] • For Fixed Matrices$R, S$, the Matrices$RAS$form a Subspace Let$V$be the vector space of$k \times k$matrices. Then for fixed matrices$R, S \in V$, define the subset$W = \{ R A S \mid A \in V \}$. Prove that$W$is a vector subspace of$V$. Proof. We verify the subspace criteria: the zero vector of$V$is in$W$, and […] • Calculate$A^{10}$for a Given Matrix$A$Find$A^{10}$, where$A=\begin{bmatrix} 4 & 3 & 0 & 0 \\ 3 &-4 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 \end{bmatrix}$. (Harvard University Exam) Solution. Let$B=\begin{bmatrix} 4 & 3\\ 3& -4 \end{bmatrix}$[…] • Group of Invertible Matrices Over a Finite Field and its Stabilizer Let$\F_p$be the finite field of$p$elements, where$p$is a prime number. Let$G_n=\GL_n(\F_p)$be the group of$n\times n$invertible matrices with entries in the field$\F_p$. As usual in linear algebra, we may regard the elements of$G_n$as linear transformations on$\F_p^n$, […] • Rotation Matrix in Space and its Determinant and Eigenvalues For a real number$0\leq \theta \leq \pi$, we define the real$3\times 3$matrix$A$by $A=\begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta &\cos\theta &0 \\ 0 & 0 & 1 \end{bmatrix}.$ (a) Find the determinant of the matrix$A$. (b) Show that$A$is an […] • Vector Space of Polynomials and Coordinate Vectors Let$P_2$be the vector space of all polynomials of degree two or less. Consider the subset in$P_2$Q=\{ p_1(x), p_2(x), p_3(x), p_4(x)\},$ where \begin{align*} &p_1(x)=x^2+2x+1, &p_2(x)=2x^2+3x+1, \\ &p_3(x)=2x^2, &p_4(x)=2x^2+x+1. \end{align*} (a) Use the basis […] • Inner Products, Lengths, and Distances of 3-Dimensional Real Vectors For this problem, use the real vectors $\mathbf{v}_1 = \begin{bmatrix} -1 \\ 0 \\ 2 \end{bmatrix} , \mathbf{v}_2 = \begin{bmatrix} 0 \\ 2 \\ -3 \end{bmatrix} , \mathbf{v}_3 = \begin{bmatrix} 2 \\ 2 \\ 3 \end{bmatrix} .$ Suppose that\mathbf{v}_4$is another vector which is […] • Diagonalizable Matrix with Eigenvalue 1, -1 Suppose that$A$is a diagonalizable$n\times n$matrix and has only$1$and$-1$as eigenvalues. Show that$A^2=I_n$, where$I_n$is the$n\times n\$ identity matrix. (Stanford University Linear Algebra Exam) See below for a generalized problem. Hint. Diagonalize the […]