Author: Yu

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)
 

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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)

 

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A 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.

 

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Differentiation 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).

 

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Finitely 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.

 

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Restriction 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.

 

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If $A$ 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.

 

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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.

 

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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 $A=B$.

 

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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 $\phi$?

 

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