# mmc%20rep%20of%20mcg

by Yu ·

Add to solve later

Sponsored Links

Add to solve later

Sponsored Links

Add to solve later

Sponsored Links

### More from my site

- Express a Hermitian Matrix as a Sum of Real Symmetric Matrix and a Real Skew-Symmetric Matrix Recall that a complex matrix is called Hermitian if $A^*=A$, where $A^*=\bar{A}^{\trans}$. Prove that every Hermitian matrix $A$ can be written as the sum \[A=B+iC,\] where $B$ is a real symmetric matrix and $C$ is a real skew-symmetric matrix. Proof. Since […]
- Prove that any Algebraic Closed Field is Infinite Prove that any algebraic closed field is infinite. Definition. A field $F$ is said to be algebraically closed if each non-constant polynomial in $F[x]$ has a root in $F$. Proof. Let $F$ be a finite field and consider the polynomial \[f(x)=1+\prod_{a\in […]
- Summary: Possibilities for the Solution Set of a System of Linear Equations In this post, we summarize theorems about the possibilities for the solution set of a system of linear equations and solve the following problems. Determine all possibilities for the solution set of the system of linear equations described below. (a) A homogeneous system of $3$ […]
- Rotation Matrix in Space and its Determinant and Eigenvalues For a real number $0\leq \theta \leq \pi$, we define the real $3\times 3$ matrix $A$ by \[A=\begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta &\cos\theta &0 \\ 0 & 0 & 1 \end{bmatrix}.\] (a) Find the determinant of the matrix $A$. (b) Show that $A$ is an […]
- Find the Inverse Matrices if Matrices are Invertible by Elementary Row Operations For each of the following $3\times 3$ matrices $A$, determine whether $A$ is invertible and find the inverse $A^{-1}$ if exists by computing the augmented matrix $[A|I]$, where $I$ is the $3\times 3$ identity matrix. (a) $A=\begin{bmatrix} 1 & 3 & -2 \\ 2 &3 &0 \\ […]
- Row Equivalent Matrix, Bases for the Null Space, Range, and Row Space of a Matrix Let \[A=\begin{bmatrix} 1 & 1 & 2 \\ 2 &2 &4 \\ 2 & 3 & 5 \end{bmatrix}.\] (a) Find a matrix $B$ in reduced row echelon form such that $B$ is row equivalent to the matrix $A$. (b) Find a basis for the null space of $A$. (c) Find a basis for the range of $A$ that […]
- Every Plane Through the Origin in the Three Dimensional Space is a Subspace Prove that every plane in the $3$-dimensional space $\R^3$ that passes through the origin is a subspace of $\R^3$. Proof. Each plane $P$ in $\R^3$ through the origin is given by the equation \[ax+by+cz=0\] for some real numbers $a, b, c$. That is, the […]
- Quiz 3. Condition that Vectors are Linearly Dependent/ Orthogonal Vectors are Linearly Independent (a) For what value(s) of $a$ is the following set $S$ linearly dependent? \[ S=\left \{\,\begin{bmatrix} 1 \\ 2 \\ 3 \\ a \end{bmatrix}, \begin{bmatrix} a \\ 0 \\ -1 \\ 2 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ a^2 […]