The Group of Rational Numbers is Not Finitely Generated
(a) Prove that the additive group $\Q=(\Q, +)$ of rational numbers is not finitely generated.
(b) Prove that the multiplicative group $\Q^*=(\Q\setminus\{0\}, \times)$ of nonzero rational numbers is not finitely generated.
Every Finitely Generated Subgroup of Additive Group $\Q$ of Rational Numbers is Cyclic
Let $\Q=(\Q, +)$ be the additive group of rational numbers.
(a) Prove that every finitely generated subgroup of $(\Q, +)$ is cyclic.
(b) Prove that $\Q$ and $\Q \times \Q$ are not isomorphic as groups.
Determine the Values of $a$ such that the 2 by 2 Matrix is Diagonalizable
\[A=\begin{bmatrix}
1-a & a\\
-a& 1+a
\end{bmatrix}\] be a $2\times 2$ matrix, where $a$ is a complex number.
Determine the values of $a$ such that the matrix $A$ is diagonalizable.
(Nagoya University, Linear Algebra Exam Problem)
Prove that a Group of Order 217 is Cyclic and Find the Number of Generators
Let $G$ be a finite group of order $217$.
(a) Prove that $G$ is a cyclic group.
(b) Determine the number of generators of the group $G$.
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
\[A\mathbf{x}+\epsilon\mathbf{x}=\mathbf{v}\] has a unique solution $\mathbf{x}=\mathbf{x}(\epsilon) \in \R^n$.
(b) Evaluate
\[\lim_{\epsilon \to 0^+} \epsilon \mathbf{x}(\epsilon)\] in terms of $\mathbf{v}$, the eigenvectors of $A$, and the inner product $\langle\, ,\,\rangle$ on $\R^n$.
(University of California, Berkeley, Linear Algebra Qualifying Exam)
Diagonalize the 3 by 3 Matrix if it is Diagonalizable
Determine whether the matrix
0 & 1 & 0 \\
-1 &0 &0 \\
0 & 0 & 2
\end{bmatrix}\] is diagonalizable.
If it is diagonalizable, then find the invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.
The Order of a Conjugacy Class Divides the Order of the Group
Let $G$ be a finite group.
The centralizer of an element $a$ of $G$ is defined to be
\[C_G(a)=\{g\in G \mid ga=ag\}.\]
A conjugacy class is a set of the form
\[\Cl(a)=\{bab^{-1} \mid b\in G\}\] for some $a\in G$.
(a) Prove that the centralizer of an element of $a$ in $G$ is a subgroup of the group $G$.
(b) Prove that the order (the number of elements) of every conjugacy class in $G$ divides the order of the group $G$.
All Linear Transformations that Take the Line $y=x$ to the Line $y=-x$
Determine all linear transformations of the $2$-dimensional $x$-$y$ plane $\R^2$ that take the line $y=x$ to the line $y=-x$.
Differentiating Linear Transformation is Nilpotent
Let $P_n$ be the vector space of all polynomials with real coefficients of degree $n$ or less.
Consider the differentiation linear transformation $T: P_n\to P_n$ defined by
\[T\left(\, f(x) \,\right)=\frac{d}{dx}f(x).\]
(a) Consider the case $n=2$. Let $B=\{1, x, x^2\}$ be a basis of $P_2$. Find the matrix representation $A$ of the linear transformation $T$ with respect to the basis $B$.
(b) Compute $A^3$, where $A$ is the matrix obtained in part (a).
(c) If you computed $A^3$ in part (b) directly, then is there any theoretical explanation of your result?
(d) Now we consider the general case. Let $B$ be any basis of the vector space of $P_n$ and let $A$ be the matrix representation of the linear transformation $T$ with respect to the basis $B$.
Prove that without any calculation that the matrix $A$ is nilpotent.
Eigenvalues of Similarity Transformations
Let $A$ be an $n\times n$ complex matrix.
Let $S$ be an invertible matrix.
(a) If $SAS^{-1}=\lambda A$ for some complex number $\lambda$, then prove that either $\lambda^n=1$ or $A$ is a singular matrix.
(b) If $n$ is odd and $SAS^{-1}=-A$, then prove that $0$ is an eigenvalue of $A$.
(c) Suppose that all the eigenvalues of $A$ are integers and $\det(A) > 0$. If $n$ is odd and $SAS^{-1}=A^{-1}$, then prove that $1$ is an eigenvalue of $A$.
Inequality about Eigenvalue of a Real Symmetric Matrix
Let $A$ be an $n\times n$ real symmetric matrix.
Prove that there exists an eigenvalue $\lambda$ of $A$ such that for any vector $\mathbf{v}\in \R^n$, we have the inequality
\[\mathbf{v}\cdot A\mathbf{v} \leq \lambda \|\mathbf{v}\|^2.\]
Null Space, Nullity, Range, Rank of a Projection Linear Transformation
Let $\mathbf{u}=\begin{bmatrix}
1 \\
\end{bmatrix}$ and $T:\R^3 \to \R^3$ be the linear transformation
\[T(\mathbf{x})=\proj_{\mathbf{u}}\mathbf{x}=\left(\, \frac{\mathbf{u}\cdot \mathbf{x}}{\mathbf{u}\cdot \mathbf{u}} \,\right)\mathbf{u}.\]
(a) Calculate the null space $\calN(T)$, a basis for $\calN(T)$ and nullity of $T$.
(b) Only by using part (a) and no other calculations, find $\det(A)$, where $A$ is the matrix representation of $T$ with respect to the standard basis of $\R^3$.
(c) Calculate the range $\calR(T)$, a basis for $\calR(T)$ and the rank of $T$.
(d) Calculate the matrix $A$ representing $T$ with respect to the standard basis for $\R^3$.
(e) Let
\[B=\left\{\, \begin{bmatrix}
\end{bmatrix}, \begin{bmatrix}
-1 \\
\end{bmatrix} \,\right\}\] be a basis for $\R^3$.
Calculate the coordinates of $\begin{bmatrix}
x \\
y \\
\end{bmatrix}$ with respect to $B$.
(The Ohio State University, Linear Algebra Exam Problem)
A Module $M$ is Irreducible if and only if $M$ is isomorphic to $R/I$ for a Maximal Ideal $I$.
Let $R$ be a commutative ring with $1$ and let $M$ be an $R$-module.
Prove that the $R$-module $M$ is irreducible if and only if $M$ is isomorphic to $R/I$, where $I$ is a maximal ideal of $R$, as an $R$-module.
The Product of a Subgroup and a Normal Subgroup is a Subgroup
Let $G$ be a group. Let $H$ be a subgroup of $G$ and let $N$ be a normal subgroup of $G$.
The product of $H$ and $N$ is defined to be the subset
\[H\cdot N=\{hn\in G\mid h \in H, n\in N\}.\] Prove that the product $H\cdot N$ is a subgroup of $G$.
If $A^{\trans}A=A$, then $A$ is a Symmetric Idempotent Matrix
Let $A$ be a square matrix such that
\[A^{\trans}A=A,\] where $A^{\trans}$ is the transpose matrix of $A$.
Prove that $A$ is idempotent, that is, $A^2=A$. Also, prove that $A$ is a symmetric matrix.
A One-Line Proof that there are Infinitely Many Prime Numbers
Prove that there are infinitely many prime numbers in ONE-LINE.
Inverse Map of a Bijective Homomorphism is a Group Homomorphism
Let $G$ and $H$ be groups and let $\phi: G \to H$ be a group homomorphism.
Suppose that $f:G\to H$ is bijective.
Then there exists a map $\psi:H\to G$ such that
\[\psi \circ \phi=\id_G \text{ and } \phi \circ \psi=\id_H.\] Then prove that $\psi:H \to G$ is also a group homomorphism.
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
\[\phi(g^{-1})=\phi(g)^{-1}.\]
Injective Group Homomorphism that does not have Inverse Homomorphism
Let $A=B=\Z$ be the additive group of integers.
Define a map $\phi: A\to B$ by sending $n$ to $2n$ for any integer $n\in A$.
(a) Prove that $\phi$ is a group homomorphism.
(b) Prove that $\phi$ is injective.
(c) Prove that there does not exist a group homomorphism $\psi:B \to A$ such that $\psi \circ \phi=\id_A$.
Solve the System of Linear Equations Using the Inverse Matrix of the Coefficient Matrix
Consider the following system of linear equations
\begin{align*}
2x+3y+z&=-1\\
3x+3y+z&=1\\
2x+4y+z&=-2.
\end{align*}
(a) Find the coefficient matrix $A$ for this system.
(b) Find the inverse matrix of the coefficient matrix found in (a)
(c) Solve the system using the inverse matrix $A^{-1}$.
Page 15 of 37« First«...1213141516171819...30...»Last »
Are Coefficient Matrices of the Systems of Linear Equations Nonsingular?
Idempotent Matrix and its Eigenvalues
Every Cyclic Group is Abelian
Quiz 1. Gauss-Jordan Elimination / Homogeneous System. Math 2568 Spring 2017.
Find a Basis for Nullspace, Row Space, and Range of a Matrix