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 […]
Sequence Converges to the Largest Eigenvalue of a Matrix
Let $A$ be an $n\times n$ matrix. Suppose that $A$ has real eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_n$ with corresponding eigenvectors $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$.
Furthermore, suppose that
\[|\lambda_1| > |\lambda_2| \geq \cdots \geq […]
Group Homomorphism Sends the Inverse Element to the Inverse Element
Let $G, G'$ be groups. Let $\phi:G\to G'$ be a group homomorphism.
Then prove that for any element $g\in G$, we have
\[\phi(g^{-1})=\phi(g)^{-1}.\]
Definition (Group homomorphism).
A map $\phi:G\to G'$ is called a group homomorphism […]
A Matrix is Invertible If and Only If It is Nonsingular
In this problem, we will show that the concept of non-singularity of a matrix is equivalent to the concept of invertibility.
That is, we will prove that:
A matrix $A$ is nonsingular if and only if $A$ is invertible.
(a) Show that if $A$ is invertible, then $A$ is […]
A Condition that a Linear System has Nontrivial Solutions
For what value(s) of $a$ does the system have nontrivial solutions?
\begin{align*}
&x_1+2x_2+x_3=0\\
&-x_1-x_2+x_3=0\\
& 3x_1+4x_2+ax_3=0.
\end{align*}
Solution.
First note that the system is homogeneous and hence it is consistent. Thus if the system has a nontrivial […]
Determinant/Trace and Eigenvalues of a Matrix
Let $A$ be an $n\times n$ matrix and let $\lambda_1, \dots, \lambda_n$ be its eigenvalues.
Show that
(1) $$\det(A)=\prod_{i=1}^n \lambda_i$$
(2) $$\tr(A)=\sum_{i=1}^n \lambda_i$$
Here $\det(A)$ is the determinant of the matrix $A$ and $\tr(A)$ is the trace of the matrix […]
Are these vectors in the Nullspace of the Matrix?
Let $A=\begin{bmatrix}
1 & 0 & 3 & -2 \\
0 &3 & 1 & 1 \\
1 & 3 & 4 & -1
\end{bmatrix}$. For each of the following vectors, determine whether the vector is in the nullspace $\calN(A)$.
(a) $\begin{bmatrix}
-3 \\
0 \\
1 \\
0
\end{bmatrix}$
[…]
Possibilities For the Number of Solutions for a Linear System
Determine whether the following systems of equations (or matrix equations) described below has no solution, one unique solution or infinitely many solutions and justify your answer.
(a) \[\left\{
\begin{array}{c}
ax+by=c \\
dx+ey=f,
\end{array}
\right.
\]
where $a,b,c, d$ […]