Prove that any Set of Vectors Containing the Zero Vector is Linearly Dependent
Prove that any set of vectors which contains the zero vector is linearly dependent.
Solution.
Let $\mathbf{0}$ be the zero vector, and $\mathbf{v}_1, \cdots, \mathbf{v}_k$ are the other vectors in the set.
Then we have the non-trivial linear combination
\[1 \cdot […]
How to Find a Basis for the 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) […]
Every Integral Domain Artinian Ring is a Field
Let $R$ be a ring with $1$. Suppose that $R$ is an integral domain and an Artinian ring.
Prove that $R$ is a field.
Definition (Artinian ring).
A ring $R$ is called Artinian if it satisfies the defending chain condition on ideals.
That is, whenever we have […]
Solving a System of Linear Equations Using Gaussian Elimination
Solve the following system of linear equations using Gaussian elimination.
\begin{align*}
x+2y+3z &=4 \\
5x+6y+7z &=8\\
9x+10y+11z &=12
\end{align*}
Elementary row operations
The three elementary row operations on a matrix are defined as […]
Determine the Number of Elements of Order 3 in a Non-Cyclic Group of Order 57
Let $G$ be a group of order $57$. Assume that $G$ is not a cyclic group.
Then determine the number of elements in $G$ of order $3$.
Proof.
Observe the prime factorization $57=3\cdot 19$.
Let $n_{19}$ be the number of Sylow $19$-subgroups of $G$.
By […]
Isomorphism Criterion of Semidirect Product of Groups
Let $A$, $B$ be groups. Let $\phi:B \to \Aut(A)$ be a group homomorphism.
The semidirect product $A \rtimes_{\phi} B$ with respect to $\phi$ is a group whose underlying set is $A \times B$ with group operation
\[(a_1, b_1)\cdot (a_2, b_2)=(a_1\phi(b_1)(a_2), b_1b_2),\]
where $a_i […]
A Positive Definite Matrix Has a Unique Positive Definite Square Root
Prove that a positive definite matrix has a unique positive definite square root.
In this post, we review several definitions (a square root of a matrix, a positive definite matrix) and solve the above problem.
After the proof, several extra problems about square […]
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.
[…]