# linear-algebra-eyecatch

• Simple Commutative Relation on Matrices Let $A$ and $B$ are $n \times n$ matrices with real entries. Assume that $A+B$ is invertible. Then show that $A(A+B)^{-1}B=B(A+B)^{-1}A.$ (University of California, Berkeley Qualifying Exam) Proof. Let $P=A+B$. Then $B=P-A$. Using these, we express the given […]
• Determine the Dimension of a Mysterious Vector Space From Coordinate Vectors Let $V$ be a vector space and $B$ be a basis for $V$. Let $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5$ be vectors in $V$. Suppose that $A$ is the matrix whose columns are the coordinate vectors of $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, […] • Null Space, Nullity, Range, Rank of a Projection Linear Transformation Let$\mathbf{u}=\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}$and$T:\R^3 \to \R^3$be the linear transformation $T(\mathbf{x})=\proj_{\mathbf{u}}\mathbf{x}=\left(\, \frac{\mathbf{u}\cdot \mathbf{x}}{\mathbf{u}\cdot \mathbf{u}} \,\right)\mathbf{u}.$ (a) […] • A Module is Irreducible if and only if It is a Cyclic Module With Any Nonzero Element as Generator Let$R$be a ring with$1$. A nonzero$R$-module$M$is called irreducible if$0$and$M$are the only submodules of$M$. (It is also called a simple module.) (a) Prove that a nonzero$R$-module$M$is irreducible if and only if$M$is a cyclic module with any nonzero element […] • Group of Order$pq$is Either Abelian or the Center is Trivial Let$G$be a group of order$|G|=pq$, where$p$and$q$are (not necessarily distinct) prime numbers. Then show that$G$is either abelian group or the center$Z(G)=1$. Hint. Use the result of the problem "If the Quotient by the Center is Cyclic, then the Group is […] • 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 = […]
• Does an Extra Vector Change the Span? Suppose that a set of vectors $S_1=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is a spanning set of a subspace $V$ in $\R^5$. If $\mathbf{v}_4$ is another vector in $V$, then is the set $S_2=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4\}$ still a spanning set for […]
• Unit Vectors and Idempotent Matrices A square matrix $A$ is called idempotent if $A^2=A$. (a) Let $\mathbf{u}$ be a vector in $\R^n$ with length $1$. Define the matrix $P$ to be $P=\mathbf{u}\mathbf{u}^{\trans}$. Prove that $P$ is an idempotent matrix. (b) Suppose that $\mathbf{u}$ and $\mathbf{v}$ be […]