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• Example of a Nilpotent Matrix $A$ such that $A^2\neq O$ but $A^3=O$. Find a nonzero $3\times 3$ matrix $A$ such that $A^2\neq O$ and $A^3=O$, where $O$ is the $3\times 3$ zero matrix. (Such a matrix is an example of a nilpotent matrix. See the comment after the solution.)   Solution. For example, let $A$ be the following $3\times […] • Isomorphism of the Endomorphism and the Tensor Product of a Vector Space Let$V$be a finite dimensional vector space over a field$K$and let$\End (V)$be the vector space of linear transformations from$V$to$V$. Let$\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n$be a basis for$V$. Show that the map$\phi:\End (V) \to V^{\oplus n}$defined by […] • A Recursive Relationship for a Power of a Matrix Suppose that the$2 \times 2$matrix$A$has eigenvalues$4$and$-2$. For each integer$n \geq 1$, there are real numbers$b_n , c_n$which satisfy the relation $A^{n} = b_n A + c_n I ,$ where$I$is the identity matrix. Find$b_n$and$c_n$for$2 \leq n \leq 5$, and […] • Determine the Values of$a$such that the 2 by 2 Matrix is Diagonalizable Let $A=\begin{bmatrix} 1-a & a\\ -a& 1+a \end{bmatrix}$ be a$2\times 2$matrix, where$a$is a complex number. Determine the values of$a$such that the matrix$A$is diagonalizable. (Nagoya University, Linear Algebra Exam Problem) Proof. To […] • Inner Product, Norm, and Orthogonal Vectors Let$\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3$are vectors in$\R^n$. Suppose that vectors$\mathbf{u}_1$,$\mathbf{u}_2$are orthogonal and the norm of$\mathbf{u}_2$is$4$and$\mathbf{u}_2^{\trans}\mathbf{u}_3=7$. Find the value of the real number$a$in […] • Find Values of$h$so that the Given Vectors are Linearly Independent Find the value(s) of$h$for which the following set of vectors $\left \{ \mathbf{v}_1=\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix} h \\ 1 \\ -h \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix} 1 \\ 2h \\ 3h+1 […] • Quiz 5: Example and Non-Example of Subspaces in 3-Dimensional Space Problem 1 Let W be the subset of the 3-dimensional vector space \R^3 defined by \[W=\left\{ \mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\in \R^3 \quad \middle| \quad 2x_1x_2=x_3 \right\}.$ (a) Which of the following vectors are in the subset […] • 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 […]