# UC-Berkeley-eye-catch

• Every Diagonalizable Nilpotent Matrix is the Zero Matrix Prove that if $A$ is a diagonalizable nilpotent matrix, then $A$ is the zero matrix $O$.   Definition (Nilpotent Matrix) A square matrix $A$ is called nilpotent if there exists a positive integer $k$ such that $A^k=O$. Proof. Main Part Since $A$ is […]
• Dual Vector Space and Dual Basis, Some Equality Let $V$ be a finite dimensional vector space over a field $k$ and let $V^*=\Hom(V, k)$ be the dual vector space of $V$. Let $\{v_i\}_{i=1}^n$ be a basis of $V$ and let $\{v^i\}_{i=1}^n$ be the dual basis of $V^*$. Then prove that $x=\sum_{i=1}^nv^i(x)v_i$ for any vector $x\in […] • Determine When the Given Matrix Invertible For which choice(s) of the constant$k$is the following matrix invertible? $A=\begin{bmatrix} 1 & 1 & 1 \\ 1 &2 &k \\ 1 & 4 & k^2 \end{bmatrix}.$ (Johns Hopkins University, Linear Algebra Exam) Hint. An$n\times n$matrix is […] • Construction of a Symmetric Matrix whose Inverse Matrix is Itself Let$\mathbf{v}$be a nonzero vector in$\R^n$. Then the dot product$\mathbf{v}\cdot \mathbf{v}=\mathbf{v}^{\trans}\mathbf{v}\neq 0$. Set$a:=\frac{2}{\mathbf{v}^{\trans}\mathbf{v}}$and define the$n\times n$matrix$A$by $A=I-a\mathbf{v}\mathbf{v}^{\trans},$ where […] • If the Matrix Product$AB=0$, then is$BA=0$as Well? Let$A$and$B$be$n\times n$matrices. Suppose that the matrix product$AB=O$, where$O$is the$n\times n$zero matrix. Is it true that the matrix product with opposite order$BA$is also the zero matrix? If so, give a proof. If not, give a […] • Non-Example of a Subspace in 3-dimensional Vector Space$\R^3$Let$S$be the following subset of the 3-dimensional vector space$\R^3$. $S=\left\{ \mathbf{x}\in \R^3 \quad \middle| \quad \mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}, x_1, x_2, x_3 \in \Z \right\},$ where$\Z$is the set of all integers. […] • Rotation Matrix in the Plane and its Eigenvalues and Eigenvectors Consider the$2\times 2$matrix $A=\begin{bmatrix} \cos \theta & -\sin \theta\\ \sin \theta& \cos \theta \end{bmatrix},$ where$\theta$is a real number$0\leq \theta < 2\pi$. (a) Find the characteristic polynomial of the matrix$A$. (b) Find the […] • The Range and Null Space of the Zero Transformation of Vector Spaces Let$U$and$V$be vector spaces over a scalar field$\F$. Define the map$T:U\to V$by$T(\mathbf{u})=\mathbf{0}_V$for each vector$\mathbf{u}\in U$. (a) Prove that$T:U\to V$is a linear transformation. (Hence,$T\$ is called the zero transformation.) (b) Determine […]