A Matrix Equation of a Symmetric Matrix and the Limit of its Solution
Let $A$ be a real symmetric $n\times n$ matrix with $0$ as a simple eigenvalue (that is, the algebraic multiplicity of the eigenvalue $0$ is $1$), and let us fix a vector $\mathbf{v}\in \R^n$.
(a) Prove that for sufficiently small positive real $\epsilon$, the equation […]
The Matrix for the Linear Transformation of the Reflection Across a Line in the Plane
Let $T:\R^2 \to \R^2$ be a linear transformation of the $2$-dimensional vector space $\R^2$ (the $x$-$y$-plane) to itself which is the reflection across a line $y=mx$ for some $m\in \R$.
Then find the matrix representation of the linear transformation $T$ with respect to the […]
The Preimage of a Normal Subgroup Under a Group Homomorphism is Normal
Let $G$ and $G'$ be groups and let $f:G \to G'$ be a group homomorphism.
If $H'$ is a normal subgroup of the group $G'$, then show that $H=f^{-1}(H')$ is a normal subgroup of the group $G$.
Proof.
We prove that $H$ is normal in $G$. (The fact that $H$ is a subgroup […]
Find All Matrices Satisfying a Given Relation
Let $a$ and $b$ be two distinct positive real numbers. Define matrices
\[A:=\begin{bmatrix}
0 & a\\
a & 0
\end{bmatrix}, \,\,
B:=\begin{bmatrix}
0 & b\\
b& 0
\end{bmatrix}.\]
Find all the pairs $(\lambda, X)$, where $\lambda$ is a real number and $X$ is a […]
Abelian Groups and Surjective Group Homomorphism
Let $G, G'$ be groups. Suppose that we have a surjective group homomorphism $f:G\to G'$.
Show that if $G$ is an abelian group, then so is $G'$.
Definitions.
Recall the relevant definitions.
A group homomorphism $f:G\to G'$ is a map from $G$ to $G'$ […]
10 True of False Problems about Nonsingular / Invertible Matrices
10 questions about nonsingular matrices, invertible matrices, and linearly independent vectors.
The quiz is designed to test your understanding of the basic properties of these topics.
You can take the quiz as many times as you like.
The solutions will be given after […]
Eigenvalues and their Algebraic Multiplicities of a Matrix with a Variable
Determine all eigenvalues and their algebraic multiplicities of the matrix
\[A=\begin{bmatrix}
1 & a & 1 \\
a &1 &a \\
1 & a & 1
\end{bmatrix},\]
where $a$ is a real number.
Proof.
To find eigenvalues we first compute the characteristic polynomial of the […]
Exponential Functions are Linearly Independent
Let $c_1, c_2,\dots, c_n$ be mutually distinct real numbers.
Show that exponential functions
\[e^{c_1x}, e^{c_2x}, \dots, e^{c_nx}\]
are linearly independent over $\R$.
Hint.
Consider a linear combination \[a_1 e^{c_1 x}+a_2 e^{c_2x}+\cdots + a_ne^{c_nx}=0.\]
[…]
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A Matrix Equation of a Symmetric Matrix and the Limit of its Solution
Let $A$ be a real symmetric $n\times n$ matrix with $0$ as a simple eigenvalue (that is, the algebraic multiplicity of the eigenvalue $0$ is $1$), and let us fix a vector $\mathbf{v}\in \R^n$.
(a) Prove that for sufficiently small positive real $\epsilon$, the equation […]
The Matrix for the Linear Transformation of the Reflection Across a Line in the Plane
Let $T:\R^2 \to \R^2$ be a linear transformation of the $2$-dimensional vector space $\R^2$ (the $x$-$y$-plane) to itself which is the reflection across a line $y=mx$ for some $m\in \R$.
Then find the matrix representation of the linear transformation $T$ with respect to the […]
The Preimage of a Normal Subgroup Under a Group Homomorphism is Normal
Let $G$ and $G'$ be groups and let $f:G \to G'$ be a group homomorphism.
If $H'$ is a normal subgroup of the group $G'$, then show that $H=f^{-1}(H')$ is a normal subgroup of the group $G$.
Proof.
We prove that $H$ is normal in $G$. (The fact that $H$ is a subgroup […]
Find All Matrices Satisfying a Given Relation
Let $a$ and $b$ be two distinct positive real numbers. Define matrices
\[A:=\begin{bmatrix}
0 & a\\
a & 0
\end{bmatrix}, \,\,
B:=\begin{bmatrix}
0 & b\\
b& 0
\end{bmatrix}.\]
Find all the pairs $(\lambda, X)$, where $\lambda$ is a real number and $X$ is a […]
Abelian Groups and Surjective Group Homomorphism
Let $G, G'$ be groups. Suppose that we have a surjective group homomorphism $f:G\to G'$.
Show that if $G$ is an abelian group, then so is $G'$.
Definitions.
Recall the relevant definitions.
A group homomorphism $f:G\to G'$ is a map from $G$ to $G'$ […]
10 True of False Problems about Nonsingular / Invertible Matrices
10 questions about nonsingular matrices, invertible matrices, and linearly independent vectors.
The quiz is designed to test your understanding of the basic properties of these topics.
You can take the quiz as many times as you like.
The solutions will be given after […]
Eigenvalues and their Algebraic Multiplicities of a Matrix with a Variable
Determine all eigenvalues and their algebraic multiplicities of the matrix
\[A=\begin{bmatrix}
1 & a & 1 \\
a &1 &a \\
1 & a & 1
\end{bmatrix},\]
where $a$ is a real number.
Proof.
To find eigenvalues we first compute the characteristic polynomial of the […]
Exponential Functions are Linearly Independent
Let $c_1, c_2,\dots, c_n$ be mutually distinct real numbers.
Show that exponential functions
\[e^{c_1x}, e^{c_2x}, \dots, e^{c_nx}\]
are linearly independent over $\R$.
Hint.
Consider a linear combination \[a_1 e^{c_1 x}+a_2 e^{c_2x}+\cdots + a_ne^{c_nx}=0.\]
[…]
