True of False Problems on Determinants and Invertible Matrices
Problem 438
Determine whether each of the following statements is True or False.
(a) If $A$ and $B$ are $n \times n$ matrices, and $P$ is an invertible $n \times n$ matrix such that $A=PBP^{-1}$, then $\det(A)=\det(B)$.
(b) If the characteristic polynomial of an $n \times n$ matrix $A$ is
\[p(\lambda)=(\lambda-1)^n+2,\]
then $A$ is invertible.
(c) If $A^2$ is an invertible $n\times n$ matrix, then $A^3$ is also invertible.
(d) If $A$ is a $3\times 3$ matrix such that $\det(A)=7$, then $\det(2A^{\trans}A^{-1})=2$.
(e) If $\mathbf{v}$ is an eigenvector of an $n \times n$ matrix $A$ with corresponding eigenvalue $\lambda_1$, and if $\mathbf{w}$ is an eigenvector of $A$ with corresponding eigenvalue $\lambda_2$, then $\mathbf{v}+\mathbf{w}$ is an eigenvector of $A$ with corresponding eigenvalue $\lambda_1+\lambda_2$.
(Stanford University, Linear Algebra Exam Problem)
Read solution
Every Integral Domain Artinian Ring is a Field
Problem 437
Let $R$ be a ring with $1$. Suppose that $R$ is an integral domain and an Artinian ring.
Prove that $R$ is a field.
Three Equivalent Conditions for a Ring to be a Field
Problem 436
Let $R$ be a ring with $1$. Prove that the following three statements are equivalent.
- The ring $R$ is a field.
- The only ideals of $R$ are $(0)$ and $R$.
- Let $S$ be any ring with $1$. Then any ring homomorphism $f:R \to S$ is injective.
Subspace Spanned By Cosine and Sine Functions
Problem 435
Let $\calF[0, 2\pi]$ be the vector space of all real valued functions defined on the interval $[0, 2\pi]$.
Define the map $f:\R^2 \to \calF[0, 2\pi]$ by
\[\left(\, f\left(\, \begin{bmatrix}
\alpha \\
\beta
\end{bmatrix} \,\right) \,\right)(x):=\alpha \cos x + \beta \sin x.\]
We put
\[V:=\im f=\{\alpha \cos x + \beta \sin x \in \calF[0, 2\pi] \mid \alpha, \beta \in \R\}.\]
(a) Prove that the map $f$ is a linear transformation.
(b) Prove that the set $\{\cos x, \sin x\}$ is a basis of the vector space $V$.
(c) Prove that the kernel is trivial, that is, $\ker f=\{\mathbf{0}\}$.
(This yields an isomorphism of $\R^2$ and $V$.)
(d) Define a map $g:V \to V$ by
\[g(\alpha \cos x + \beta \sin x):=\frac{d}{dx}(\alpha \cos x+ \beta \sin x)=\beta \cos x -\alpha \sin x.\]
Prove that the map $g$ is a linear transformation.
(e) Find the matrix representation of the linear transformation $g$ with respect to the basis $\{\cos x, \sin x\}$.
(Kyoto University, Linear Algebra exam problem)
Add to solve laterA Module is Irreducible if and only if It is a Cyclic Module With Any Nonzero Element as Generator
Problem 434
Let $R$ be a ring with $1$.
A nonzero $R$-module $M$ is called irreducible if $0$ and $M$ are the only submodules of $M$.
(It is also called a simple module.)
(a) Prove that a nonzero $R$-module $M$ is irreducible if and only if $M$ is a cyclic module with any nonzero element as its generator.
(b) Determine all the irreducible $\Z$-modules.
Add to solve laterDifferentiation is a Linear Transformation
Problem 433
Let $P_3$ be the vector space of polynomials of degree $3$ or less with real coefficients.
(a) Prove that the differentiation is a linear transformation. That is, prove that the map $T:P_3 \to P_3$ defined by
\[T\left(\, f(x) \,\right)=\frac{d}{dx} f(x)\]
for any $f(x)\in P_3$ is a linear transformation.
(b) Let $B=\{1, x, x^2, x^3\}$ be a basis of $P_3$. With respect to the basis $B$, find the matrix representation of the linear transformation $T$ in part (a).
Add to solve laterFinitely Generated Torsion Module Over an Integral Domain Has a Nonzero Annihilator
Problem 432
(a) Let $R$ be an integral domain and let $M$ be a finitely generated torsion $R$-module.
Prove that the module $M$ has a nonzero annihilator.
In other words, show that there is a nonzero element $r\in R$ such that $rm=0$ for all $m\in M$.
Here $r$ does not depend on $m$.
(b) Find an example of an integral domain $R$ and a torsion $R$-module $M$ whose annihilator is the zero ideal.
Add to solve laterNilpotent Ideal and Surjective Module Homomorphisms
Problem 431
Let $R$ be a commutative ring and let $I$ be a nilpotent ideal of $R$.
Let $M$ and $N$ be $R$-modules and let $\phi:M\to N$ be an $R$-module homomorphism.
Prove that if the induced homomorphism $\bar{\phi}: M/IM \to N/IN$ is surjective, then $\phi$ is surjective.
Add to solve laterThe Sum of Subspaces is a Subspace of a Vector Space
Problem 430
Let $V$ be a vector space over a field $K$.
If $W_1$ and $W_2$ are subspaces of $V$, then prove that the subset
\[W_1+W_2:=\{\mathbf{x}+\mathbf{y} \mid \mathbf{x}\in W_1, \mathbf{y}\in W_2\}\]
is a subspace of the vector space $V$.
Idempotent Matrices are Diagonalizable
Problem 429
Let $A$ be an $n\times n$ idempotent matrix, that is, $A^2=A$. Then prove that $A$ is diagonalizable.
Add to solve laterRestriction of a Linear Transformation on the x-z Plane is a Linear Transformation
Problem 428
Let $T:\R^3 \to \R^3$ be a linear transformation and suppose that its matrix representation with respect to the standard basis is given by the matrix
\[A=\begin{bmatrix}
1 & 0 & 2 \\
0 &3 &0 \\
4 & 0 & 5
\end{bmatrix}.\]
(a) Prove that the linear transformation $T$ sends points on the $x$-$z$ plane to points on the $x$-$z$ plane.
(b) Prove that the restriction of $T$ on the $x$-$z$ plane is a linear transformation.
(c) Find the matrix representation of the linear transformation obtained in part (b) with respect to the standard basis
\[\left\{\, \begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix}, \begin{bmatrix}
0 \\
0 \\
1
\end{bmatrix} \,\right\}\]
of the $x$-$z$ plane.
Union of Subspaces is a Subspace if and only if One is Included in Another
Problem 427
Let $W_1, W_2$ be subspaces of a vector space $V$. Then prove that $W_1 \cup W_2$ is a subspace of $V$ if and only if $W_1 \subset W_2$ or $W_2 \subset W_1$.
Add to solve laterIf $A$ is an Idempotent Matrix, then When $I-kA$ is an Idempotent Matrix?
Problem 426
A square matrix $A$ is called idempotent if $A^2=A$.
(a) Suppose $A$ is an $n \times n$ idempotent matrix and let $I$ be the $n\times n$ identity matrix. Prove that the matrix $I-A$ is an idempotent matrix.
(b) Assume that $A$ is an $n\times n$ nonzero idempotent matrix. Then determine all integers $k$ such that the matrix $I-kA$ is idempotent.
(c) Let $A$ and $B$ be $n\times n$ matrices satisfying
\[AB=A \text{ and } BA=B.\]
Then prove that $A$ is an idempotent matrix.
Every Complex Matrix Can Be Written as $A=B+iC$, where $B, C$ are Hermitian Matrices
Problem 425
(a) Prove that each complex $n\times n$ matrix $A$ can be written as
\[A=B+iC,\]
where $B$ and $C$ are Hermitian matrices.
(b) Write the complex matrix
\[A=\begin{bmatrix}
i & 6\\
2-i& 1+i
\end{bmatrix}\]
as a sum $A=B+iC$, where $B$ and $C$ are Hermitian matrices.
If Two Matrices Have the Same Eigenvalues with Linearly Independent Eigenvectors, then They Are Equal
Problem 424
Let $A$ and $B$ be $n\times n$ matrices.
Suppose that $A$ and $B$ have the same eigenvalues $\lambda_1, \dots, \lambda_n$ with the same corresponding eigenvectors $\mathbf{x}_1, \dots, \mathbf{x}_n$.
Prove that if the eigenvectors $\mathbf{x}_1, \dots, \mathbf{x}_n$ are linearly independent, then $A=B$.
Determine All Matrices Satisfying Some Conditions on Eigenvalues and Eigenvectors
Problem 423
Determine all $2\times 2$ matrices $A$ such that $A$ has eigenvalues $2$ and $-1$ with corresponding eigenvectors
\[\begin{bmatrix}
1 \\
0
\end{bmatrix} \text{ and } \begin{bmatrix}
2 \\
1
\end{bmatrix},\]
respectively.
Difference Between Ring Homomorphisms and Module Homomorphisms
Problem 422
Let $R$ be a ring with $1$ and consider $R$ as a module over itself.
(a) Determine whether every module homomorphism $\phi:R\to R$ is a ring homomorphism.
(b) Determine whether every ring homomorphism $\phi: R\to R$ is a module homomorphism.
(c) If $\phi:R\to R$ is both a module homomorphism and a ring homomorphism, what can we say about $\phi$?
Add to solve laterFind the Inverse Matrix Using the Cayley-Hamilton Theorem
Problem 421
Find the inverse matrix of the matrix
\[A=\begin{bmatrix}
1 & 1 & 2 \\
9 &2 &0 \\
5 & 0 & 3
\end{bmatrix}\]
using the Cayley–Hamilton theorem.
Fundamental Theorem of Finitely Generated Abelian Groups and its application
Problem 420
In this post, we study the Fundamental Theorem of Finitely Generated Abelian Groups, and as an application we solve the following problem.
Problem.
Let $G$ be a finite abelian group of order $n$.
If $n$ is the product of distinct prime numbers, then prove that $G$ is isomorphic to the cyclic group $Z_n=\Zmod{n}$ of order $n$.