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• How to Calculate and Simplify a Matrix Polynomial Let $T=\begin{bmatrix} 1 & 0 & 2 \\ 0 &1 &1 \\ 0 & 0 & 2 \end{bmatrix}$. Calculate and simplify the expression \[-T^3+4T^2+5T-2I,\] where $I$ is the $3\times 3$ identity matrix. (The Ohio State University Linear Algebra Exam) Hint. Use the […]
• If a Matrix $A$ is Full Rank, then $\rref(A)$ is the Identity Matrix Prove that if $A$ is an $n \times n$ matrix with rank $n$, then $\rref(A)$ is the identity matrix. Here $\rref(A)$ is the matrix in reduced row echelon form that is row equivalent to the matrix $A$.   Proof. Because $A$ has rank $n$, we know that the $n \times n$ […]
• Find a Basis of the Subspace Spanned by Four Polynomials of Degree 3 or Less Let $\calP_3$ be the vector space of all polynomials of degree $3$ or less. Let \[S=\{p_1(x), p_2(x), p_3(x), p_4(x)\},\] where \begin{align*} p_1(x)&=1+3x+2x^2-x^3 & p_2(x)&=x+x^3\\ p_3(x)&=x+x^2-x^3 & p_4(x)&=3+8x+8x^3. \end{align*} (a) […]
• No Finite Abelian Group is Divisible A nontrivial abelian group $A$ is called divisible if for each element $a\in A$ and each nonzero integer $k$, there is an element $x \in A$ such that $x^k=a$. (Here the group operation of $A$ is written multiplicatively. In additive notation, the equation is written as $kx=a$.) That […]
• The Vector $S^{-1}\mathbf{v}$ is the Coordinate Vector of $\mathbf{v}$ Suppose that $B=\{\mathbf{v}_1, \mathbf{v}_2\}$ is a basis for $\R^2$. Let $S:=[\mathbf{v}_1, \mathbf{v}_2]$. Note that as the column vectors of $S$ are linearly independent, the matrix $S$ is invertible. Prove that for each vector $\mathbf{v} \in V$, the vector […]
• A Rational Root of a Monic Polynomial with Integer Coefficients is an Integer Suppose that $\alpha$ is a rational root of a monic polynomial $f(x)$ in $\Z[x]$. Prove that $\alpha$ is an integer.   Proof. Suppose that $\alpha=\frac{p}{q}$ is a rational number in lowest terms, that is, $p$ and $q$ are relatively prime […]
• Find a Basis For the Null Space of a Given $2\times 3$ Matrix Let \[A=\begin{bmatrix} 1 & 1 & 0 \\ 1 &1 &0 \end{bmatrix}\] be a matrix. Find a basis of the null space of the matrix $A$. (Remark: a null space is also called a kernel.)   Solution. The null space $\calN(A)$ of the matrix $A$ is by […]
• Every Complex Matrix Can Be Written as $A=B+iC$, where $B, C$ are Hermitian Matrices (a) Prove that each complex $n\times n$ matrix $A$ can be written as \[A=B+iC,\] where $B$ and $C$ are Hermitian matrices. (b) Write the complex matrix \[A=\begin{bmatrix} i & 6\\ 2-i& 1+i \end{bmatrix}\] as a sum $A=B+iC$, where $B$ and $C$ are Hermitian […]