# HW_frontpage

by Yu ·

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- Prove $\mathbf{x}^{\trans}A\mathbf{x} \geq 0$ and determine those $\mathbf{x}$ such that $\mathbf{x}^{\trans}A\mathbf{x}=0$ For each of the following matrix $A$, prove that $\mathbf{x}^{\trans}A\mathbf{x} \geq 0$ for all vectors $\mathbf{x}$ in $\R^2$. Also, determine those vectors $\mathbf{x}\in \R^2$ such that $\mathbf{x}^{\trans}A\mathbf{x}=0$. (a) $A=\begin{bmatrix} 4 & 2\\ 2& […]
- Compute $A^5\mathbf{u}$ Using Linear Combination Let \[A=\begin{bmatrix} -4 & -6 & -12 \\ -2 &-1 &-4 \\ 2 & 3 & 6 \end{bmatrix}, \quad \mathbf{u}=\begin{bmatrix} 6 \\ 5 \\ -3 \end{bmatrix}, \quad \mathbf{v}=\begin{bmatrix} -2 \\ 0 \\ 1 \end{bmatrix}, \quad \text{ and } […]
- Column Rank = Row Rank. (The Rank of a Matrix is the Same as the Rank of its Transpose) Let $A$ be an $m\times n$ matrix. Prove that the rank of $A$ is the same as the rank of the transpose matrix $A^{\trans}$. Hint. Recall that the rank of a matrix $A$ is the dimension of the range of $A$. The range of $A$ is spanned by the column vectors of the matrix […]
- Diagonalize the 3 by 3 Matrix Whose Entries are All One Diagonalize the matrix \[A=\begin{bmatrix} 1 & 1 & 1 \\ 1 &1 &1 \\ 1 & 1 & 1 \end{bmatrix}.\] Namely, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$. (The Ohio State University, Linear Algebra Final Exam […]
- In which $\R^k$, are the Nullspace and Range Subspaces? Let $A$ be an $m \times n$ matrix. Suppose that the nullspace of $A$ is a plane in $\R^3$ and the range is spanned by a nonzero vector $\mathbf{v}$ in $\R^5$. Determine $m$ and $n$. Also, find the rank and nullity of $A$. Solution. For an $m \times n$ matrix $A$, the […]
- Prove Vector Space Properties Using Vector Space Axioms Using the axiom of a vector space, prove the following properties. Let $V$ be a vector space over $\R$. Let $u, v, w\in V$. (a) If $u+v=u+w$, then $v=w$. (b) If $v+u=w+u$, then $v=w$. (c) The zero vector $\mathbf{0}$ is unique. (d) For each $v\in V$, the additive inverse […]
- Find a Condition that a Vector be a Linear Combination Let \[\mathbf{v}=\begin{bmatrix} a \\ b \\ c \end{bmatrix}, \qquad \mathbf{v}_1=\begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \qquad \mathbf{v}_2=\begin{bmatrix} 2 \\ -1 \\ 2 \end{bmatrix}.\] Find the necessary and […]
- Equivalent Conditions to be a Unitary Matrix A complex matrix is called unitary if $\overline{A}^{\trans} A=I$. The inner product $(\mathbf{x}, \mathbf{y})$ of complex vector $\mathbf{x}$, $\mathbf{y}$ is defined by $(\mathbf{x}, \mathbf{y}):=\overline{\mathbf{x}}^{\trans} \mathbf{y}$. The length of a complex vector […]