# linear=algebra-eye-catch2

by Yu · Published · Updated

Add to solve later

Add to solve later

Add to solve later

### More from my site

- A Condition that a Commutator Group is a Normal Subgroup Let $H$ be a normal subgroup of a group $G$. Then show that $N:=[H, G]$ is a subgroup of $H$ and $N \triangleleft G$. Here $[H, G]$ is a subgroup of $G$ generated by commutators $[h,k]:=hkh^{-1}k^{-1}$. In particular, the commutator subgroup $[G, G]$ is a normal subgroup of […]
- The Product of Distinct Sylow $p$-Subgroups Can Never be a Subgroup Let $G$ a finite group and let $H$ and $K$ be two distinct Sylow $p$-group, where $p$ is a prime number dividing the order $|G|$ of $G$. Prove that the product $HK$ can never be a subgroup of the group $G$. Hint. Use the following fact. If $H$ and $K$ […]
- If $\mathbf{v}, \mathbf{w}$ are Linearly Independent Vectors and $A$ is Nonsingular, then $A\mathbf{v}, A\mathbf{w}$ are Linearly Independent Let $A$ be an $n\times n$ nonsingular matrix. Let $\mathbf{v}, \mathbf{w}$ be linearly independent vectors in $\R^n$. Prove that the vectors $A\mathbf{v}$ and $A\mathbf{w}$ are linearly independent. Proof. Suppose that we have a linear […]
- Find a Basis of the Vector Space of Polynomials of Degree 2 or Less Among Given Polynomials Let $P_2$ be the vector space of all polynomials with real coefficients of degree $2$ or less. Let $S=\{p_1(x), p_2(x), p_3(x), p_4(x)\}$, where \begin{align*} p_1(x)&=-1+x+2x^2, \quad p_2(x)=x+3x^2\\ p_3(x)&=1+2x+8x^2, \quad p_4(x)=1+x+x^2. \end{align*} (a) Find […]
- Any Finite Group Has a Composition Series Let $G$ be a finite group. Then show that $G$ has a composition series. Proof. We prove the statement by induction on the order $|G|=n$ of the finite group. When $n=1$, this is trivial. Suppose that any finite group of order less than $n$ has a composition […]
- Find a Basis for Nullspace, Row Space, and Range of a Matrix Let $A=\begin{bmatrix} 2 & 4 & 6 & 8 \\ 1 &3 & 0 & 5 \\ 1 & 1 & 6 & 3 \end{bmatrix}$. (a) Find a basis for the nullspace of $A$. (b) Find a basis for the row space of $A$. (c) Find a basis for the range of $A$ that consists of column vectors of $A$. (d) […]
- Determine Trigonometric Functions with Given Conditions (a) Find a function \[g(\theta) = a \cos(\theta) + b \cos(2 \theta) + c \cos(3 \theta)\] such that $g(0) = g(\pi/2) = g(\pi) = 0$, where $a, b, c$ are constants. (b) Find real numbers $a, b, c$ such that the function \[g(\theta) = a \cos(\theta) + b \cos(2 \theta) + c \cos(3 […]
- 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 […]