## Every $n$-Dimensional Vector Space is Isomorphic to the Vector Space $\R^n$

## Problem 545

Let $V$ be a vector space over the field of real numbers $\R$.

Prove that if the dimension of $V$ is $n$, then $V$ is isomorphic to $\R^n$.

Add to solve laterLet $V$ be a vector space over the field of real numbers $\R$.

Prove that if the dimension of $V$ is $n$, then $V$ is isomorphic to $\R^n$.

Add to solve laterLet $G$ a finite group and let $H$ and $K$ be two distinct Sylow $p$-group, where $p$ is a prime number dividing the order $|G|$ of $G$.

Prove that the product $HK$ can never be a subgroup of the group $G$.

Add to solve later Let $R$ be a ring with $1$.

Suppose that $a, b$ are elements in $R$ such that

\[ab=1 \text{ and } ba\neq 1.\]

**(a)** Prove that $1-ba$ is idempotent.

**(b)** Prove that $b^n(1-ba)$ is nilpotent for each positive integer $n$.

**(c)** Prove that the ring $R$ has infinitely many nilpotent elements.

Let $R$ be a ring with $1\neq 0$. Let $a, b\in R$ such that $ab=1$.

**(a)** Prove that if $a$ is not a zero divisor, then $ba=1$.

**(b)** Prove that if $b$ is not a zero divisor, then $ba=1$.

Let $U$ and $V$ be finite dimensional vector spaces over a scalar field $\F$.

Consider a linear transformation $T:U\to V$.

Prove that if $\dim(U) > \dim(V)$, then $T$ cannot be injective (one-to-one).

Add to solve later Let $U$ and $V$ be vector spaces over a scalar field $\F$.

Let $T: U \to V$ be a linear transformation.

Prove that $T$ is injective (one-to-one) if and only if the nullity of $T$ is zero.

Add to solve later Consider the $2\times 2$ real matrix

\[A=\begin{bmatrix}

1 & 1\\

1& 3

\end{bmatrix}.\]

**(a)** Prove that the matrix $A$ is positive definite.

**(b)** Since $A$ is positive definite by part (a), the formula

\[\langle \mathbf{x}, \mathbf{y}\rangle:=\mathbf{x}^{\trans} A \mathbf{y}\]
for $\mathbf{x}, \mathbf{y} \in \R^2$ defines an inner product on $\R^n$.

Consider $\R^2$ as an inner product space with this inner product.

Prove that the unit vectors

\[\mathbf{e}_1=\begin{bmatrix}

1 \\

0

\end{bmatrix} \text{ and } \mathbf{e}_2=\begin{bmatrix}

0 \\

1

\end{bmatrix}\]
are not orthogonal in the inner product space $\R^2$.

**(c)** Find an orthogonal basis $\{\mathbf{v}_1, \mathbf{v}_2\}$ of $\R^2$ from the basis $\{\mathbf{e}_1, \mathbf{e}_2\}$ using the Gram-Schmidt orthogonalization process.

**(a)** Suppose that $A$ is an $n\times n$ real symmetric positive definite matrix.

Prove that

\[\langle \mathbf{x}, \mathbf{y}\rangle:=\mathbf{x}^{\trans}A\mathbf{y}\]
defines an inner product on the vector space $\R^n$.

**(b)** Let $A$ be an $n\times n$ real matrix. Suppose that

\[\langle \mathbf{x}, \mathbf{y}\rangle:=\mathbf{x}^{\trans}A\mathbf{y}\]
defines an inner product on the vector space $\R^n$.

Prove that $A$ is symmetric and positive definite.

Add to solve laterLet $A$ and $B$ be $2\times 2$ matrices such that $(AB)^2=O$, where $O$ is the $2\times 2$ zero matrix.

Determine whether $(BA)^2$ must be $O$ as well. If so, prove it. If not, give a counter example.

Add to solve laterLet $R$ and $S$ be rings with $1\neq 0$.

Prove that every ideal of the direct product $R\times S$ is of the form $I\times J$, where $I$ is an ideal of $R$, and $J$ is an ideal of $S$.

Add to solve later**(a)** Prove that every prime ideal of a Principal Ideal Domain (PID) is a maximal ideal.

**(b)** Prove that a quotient ring of a PID by a prime ideal is a PID.

Let $I$ be a nonzero ideal of the ring of Gaussian integers $\Z[i]$.

Prove that the quotient ring $\Z[i]/I$ is finite.

Add to solve later Consider the complex matrix

\[A=\begin{bmatrix}

\sqrt{2}\cos x & i \sin x & 0 \\

i \sin x &0 &-i \sin x \\

0 & -i \sin x & -\sqrt{2} \cos x

\end{bmatrix},\]
where $x$ is a real number between $0$ and $2\pi$.

Determine for which values of $x$ the matrix $A$ is diagonalizable.

When $A$ is diagonalizable, find a diagonal matrix $D$ so that $P^{-1}AP=D$ for some nonsingular matrix $P$.

Let $R$ and $S$ be rings. Suppose that $f: R \to S$ is a surjective ring homomorphism.

Prove that every image of an ideal of $R$ under $f$ is an ideal of $S$.

Namely, prove that if $I$ is an ideal of $R$, then $J=f(I)$ is an ideal of $S$.

**(a)** Let $F$ be a field. Show that $F$ does not have a nonzero zero divisor.

**(b)** Let $R$ and $S$ be nonzero rings with identities.

Prove that the direct product $R\times S$ cannot be a field.

Let $R$ be a commutative ring with identity $1\neq 0$. Suppose that for each element $a\in R$, there exists an integer $n > 1$ depending on $a$.

Then prove that every prime ideal is a maximal ideal.

Add to solve later Let $\F_3=\Zmod{3}$ be the finite field of order $3$.

Consider the ring $\F_3[x]$ of polynomial over $\F_3$ and its ideal $I=(x^2+1)$ generated by $x^2+1\in \F_3[x]$.

**(a)** Prove that the quotient ring $\F_3[x]/(x^2+1)$ is a field. How many elements does the field have?

**(b)** Let $ax+b+I$ be a nonzero element of the field $\F_3[x]/(x^2+1)$, where $a, b \in \F_3$. Find the inverse of $ax+b+I$.

**(c)** Recall that the multiplicative group of nonzero elements of a field is a cyclic group.

Confirm that the element $x$ is not a generator of $E^{\times}$, where $E=\F_3[x]/(x^2+1)$ but $x+1$ is a generator.

Add to solve later Let $V$ denote the vector space of all real $2\times 2$ matrices.

Suppose that the linear transformation from $V$ to $V$ is given as below.

\[T(A)=\begin{bmatrix}

2 & 3\\

5 & 7

\end{bmatrix}A-A\begin{bmatrix}

2 & 3\\

5 & 7

\end{bmatrix}.\]
Prove or disprove that the linear transformation $T:V\to V$ is an isomorphism.

A square matrix $A$ is called **idempotent** if $A^2=A$.

**(a)** Let $\mathbf{u}$ be a vector in $\R^n$ with length $1$.

Define the matrix $P$ to be $P=\mathbf{u}\mathbf{u}^{\trans}$.

Prove that $P$ is an idempotent matrix.

**(b)** Suppose that $\mathbf{u}$ and $\mathbf{v}$ be unit vectors in $\R^n$ such that $\mathbf{u}$ and $\mathbf{v}$ are orthogonal.

Let $Q=\mathbf{u}\mathbf{u}^{\trans}+\mathbf{v}\mathbf{v}^{\trans}$.

Prove that $Q$ is an idempotent matrix.

**(c)** Prove that each nonzero vector of the form $a\mathbf{u}+b\mathbf{v}$ for some $a, b\in \R$ is an eigenvector corresponding to the eigenvalue $1$ for the matrix $Q$ in part (b).

A ring is called **local** if it has a unique maximal ideal.

**(a)** Prove that a ring $R$ with $1$ is local if and only if the set of non-unit elements of $R$ is an ideal of $R$.

**(b)** Let $R$ be a ring with $1$ and suppose that $M$ is a maximal ideal of $R$.

Prove that if every element of $1+M$ is a unit, then $R$ is a local ring.