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- Show that the Given 2 by 2 Matrix is Singular
Consider the matrix $M = \begin{bmatrix} 1 & 4 \\ 3 & 12 \end{bmatrix}$.
(a) Show that $M$ is singular.
(b) Find a non-zero vector $\mathbf{v}$ such that $M \mathbf{v} = \mathbf{0}$, where $\mathbf{0}$ is the $2$-dimensional zero vector.
Solution.
(a) Show […]
- The Zero is the only Nilpotent Element of the Quotient Ring by its Nilradical
Prove that if $R$ is a commutative ring and $\frakN(R)$ is its nilradical, then the zero is the only nilpotent element of $R/\frakN(R)$. That is, show that $\frakN(R/\frakN(R))=0$.
Proof.
Let $r\in R$ and if $x:=r+\frakN(R) \in R/\frakN(R)$ is a nilpotent element of […]
- Give a Formula For a Linear Transformation From $\R^2$ to $\R^3$
Let $\{\mathbf{v}_1, \mathbf{v}_2\}$ be a basis of the vector space $\R^2$, where
\[\mathbf{v}_1=\begin{bmatrix}
1 \\
1
\end{bmatrix} \text{ and } \mathbf{v}_2=\begin{bmatrix}
1 \\
-1
\end{bmatrix}.\]
The action of a linear transformation $T:\R^2\to \R^3$ on the […]
- Eigenvalues and Algebraic/Geometric Multiplicities of Matrix $A+cI$
Let $A$ be an $n \times n$ matrix and let $c$ be a complex number.
(a) For each eigenvalue $\lambda$ of $A$, prove that $\lambda+c$ is an eigenvalue of the matrix $A+cI$, where $I$ is the identity matrix. What can you say about the eigenvectors corresponding to […]
- Prove that $(A + B) \mathbf{v} = A\mathbf{v} + B\mathbf{v}$ Using the Matrix Components
Let $A$ and $B$ be $n \times n$ matrices, and $\mathbf{v}$ an $n \times 1$ column vector.
Use the matrix components to prove that $(A + B) \mathbf{v} = A\mathbf{v} + B\mathbf{v}$.
Solution.
We will use the matrix components $A = (a_{i j})_{1 \leq i, j \leq n}$, $B = […]
- Group of Invertible Matrices Over a Finite Field and its Stabilizer
Let $\F_p$ be the finite field of $p$ elements, where $p$ is a prime number.
Let $G_n=\GL_n(\F_p)$ be the group of $n\times n$ invertible matrices with entries in the field $\F_p$. As usual in linear algebra, we may regard the elements of $G_n$ as linear transformations on $\F_p^n$, […]
- Solving a System of Differential Equation by Finding Eigenvalues and Eigenvectors
Consider the system of differential equations
\begin{align*}
\frac{\mathrm{d} x_1(t)}{\mathrm{d}t} & = 2 x_1(t) -x_2(t) -x_3(t)\\
\frac{\mathrm{d}x_2(t)}{\mathrm{d}t} & = -x_1(t)+2x_2(t) -x_3(t)\\
\frac{\mathrm{d}x_3(t)}{\mathrm{d}t} & = -x_1(t) -x_2(t) […]
- 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 […]