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Linear Algebra exam problems and solutions at University of California, Berkeley


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  • Every Diagonalizable Nilpotent Matrix is the Zero MatrixEvery 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 EqualityDual 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 InvertibleDetermine 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 ItselfConstruction 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?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 $\R^3$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 EigenvectorsRotation 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 SpacesThe 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 […]

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