## The Rank of the Sum of Two Matrices

## Problem 441

Let $A$ and $B$ be $m\times n$ matrices.

Prove that

\[\rk(A+B) \leq \rk(A)+\rk(B).\]

of the day

Let $A$ and $B$ be $m\times n$ matrices.

Prove that

\[\rk(A+B) \leq \rk(A)+\rk(B).\]

Let $U$ and $V$ be finite dimensional subspaces in a vector space over a scalar field $K$.

Then prove that

\[\dim(U+V) \leq \dim(U)+\dim(V).\]

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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Let $R$ be a ring with $1$. Suppose that $R$ is an integral domain and an Artinian ring.

Prove that $R$ is a field.

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.

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 later 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.

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).

**(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.

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 later 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 $V$.

Let $A$ be an $n\times n$ idempotent matrix, that is, $A^2=A$. Then prove that $A$ is diagonalizable.

Add to solve later 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.

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 laterA 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.

**(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.

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 $2\times 2$ 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.

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 $\phi$?