# Diagonalization-eye-catch

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• Are these vectors in the Nullspace of the Matrix? Let $A=\begin{bmatrix} 1 & 0 & 3 & -2 \\ 0 &3 & 1 & 1 \\ 1 & 3 & 4 & -1 \end{bmatrix}$. For each of the following vectors, determine whether the vector is in the nullspace $\calN(A)$. (a) $\begin{bmatrix} -3 \\ 0 \\ 1 \\ 0 \end{bmatrix}$ […]
• The Length of a Vector is Zero if and only if the Vector is the Zero Vector Let $\mathbf{v}$ be an $n \times 1$ column vector. Prove that $\mathbf{v}^\trans \mathbf{v} = 0$ if and only if $\mathbf{v}$ is the zero vector $\mathbf{0}$.   Proof. Let $\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}$. Then we […]
• Diagonalize a 2 by 2 Symmetric Matrix Diagonalize the $2\times 2$ matrix $A=\begin{bmatrix} 2 & -1\\ -1& 2 \end{bmatrix}$ by finding a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.   Solution. The characteristic polynomial $p(t)$ of the matrix $A$ […]
• Find All the Square Roots of a Given 2 by 2 Matrix Let $A$ be a square matrix. A matrix $B$ satisfying $B^2=A$ is call a square root of $A$. Find all the square roots of the matrix $A=\begin{bmatrix} 2 & 2\\ 2& 2 \end{bmatrix}.$   Proof. Diagonalize $A$. We first diagonalize the matrix […]
• Find a Nonsingular Matrix $A$ satisfying $3A=A^2+AB$ (a) Find a $3\times 3$ nonsingular matrix $A$ satisfying $3A=A^2+AB$, where $B=\begin{bmatrix} 2 & 0 & -1 \\ 0 &2 &-1 \\ -1 & 0 & 1 \end{bmatrix}.$ (b) Find the inverse matrix of $A$.   Solution (a) Find a $3\times 3$ nonsingular matrix $A$. Assume […]
• Determine whether the Matrix is Nonsingular from the Given Relation Let $A$ and $B$ be $3\times 3$ matrices and let $C=A-2B$. If $A\begin{bmatrix} 1 \\ 3 \\ 5 \end{bmatrix}=B\begin{bmatrix} 2 \\ 6 \\ 10 \end{bmatrix},$ then is the matrix $C$ nonsingular? If so, prove it. Otherwise, explain why not. […]
• Short Exact Sequence and Finitely Generated Modules Let $R$ be a ring with $1$. Let $0\to M\xrightarrow{f} M' \xrightarrow{g} M^{\prime\prime} \to 0 \tag{*}$ be an exact sequence of left $R$-modules. Prove that if $M$ and $M^{\prime\prime}$ are finitely generated, then $M'$ is also finitely generated.   […]
• Basis with Respect to Which the Matrix for Linear Transformation is Diagonal Let $P_1$ be the vector space of all real polynomials of degree $1$ or less. Consider the linear transformation $T: P_1 \to P_1$ defined by $T(ax+b)=(3a+b)x+a+3,$ for any $ax+b\in P_1$. (a) With respect to the basis $B=\{1, x\}$, find the matrix of the linear transformation […]