## 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 laterof the day

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 later 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 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 $V$ 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 $A^2=A$.

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

Prove that $P$ is an idempotent matrix.

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

Prove that $Q$ is an idempotent matrix.

Prove that a positive definite matrix has a unique positive definite square root.

Add to solve later Let $A$ be a square matrix. A matrix $B$ satisfying $B^2=A$ is call a **square root** of $A$.

Find all the square roots of the matrix

\[A=\begin{bmatrix}

2 & 2\\

2& 2

\end{bmatrix}.\]

**(a)** Prove that the matrix $A=\begin{bmatrix}

0 & 1\\

0& 0

\end{bmatrix}$ does not have a square root.

Namely, show that there is no complex matrix $B$ such that $B^2=A$.

**(b)** Prove that the $2\times 2$ identity matrix $I$ has infinitely many distinct square root matrices.

Using the numbers appearing in

\[\pi=3.1415926535897932384626433832795028841971693993751058209749\dots\]
we construct the matrix \[A=\begin{bmatrix}

3 & 14 &1592& 65358\\

97932& 38462643& 38& 32\\

7950& 2& 8841& 9716\\

939937510& 5820& 974& 9

\end{bmatrix}.\]

Prove that the matrix $A$ is nonsingular.

Add to solve later Let $A$ be a square matrix.

Prove that the eigenvalues of the transpose $A^{\trans}$ are the same as the eigenvalues of $A$.

Let $A$ be an $n\times n$ invertible matrix. Then prove the transpose $A^{\trans}$ is also invertible and that the inverse matrix of the transpose $A^{\trans}$ is the transpose of the inverse matrix $A^{-1}$.

Namely, show that

\[(A^{\trans})^{-1}=(A^{-1})^{\trans}.\]

Let $A$ be a singular $2\times 2$ matrix such that $\tr(A)\neq -1$ and let $I$ be the $2\times 2$ identity matrix.

Then prove that the inverse matrix of the matrix $I+A$ is given by the following formula:

\[(I+A)^{-1}=I-\frac{1}{1+\tr(A)}A.\]

Using the formula, calculate the inverse matrix of $\begin{bmatrix}

2 & 1\\

1& 2

\end{bmatrix}$.

Prove that if $A$ is a diagonalizable nilpotent matrix, then $A$ is the zero matrix $O$.

Add to solve later Find the inverse matrix of the $3\times 3$ matrix

\[A=\begin{bmatrix}

7 & 2 & -2 \\

-6 &-1 &2 \\

6 & 2 & -1

\end{bmatrix}\]
using the Cayley-Hamilton theorem.

10 questions about nonsingular matrices, invertible matrices, and linearly independent vectors.

The quiz is designed to test your understanding of the basic properties of these topics.

You can take the quiz as many times as you like.

The solutions will be given after completing all the 10 problems.

Click the **View question** button to see the solutions.

Let $T:\R^2 \to \R^2$ be a linear transformation of the $2$-dimensional vector space $\R^2$ (the $x$-$y$-plane) to itself which is the reflection across a line $y=mx$ for some $m\in \R$.

Then find the matrix representation of the linear transformation $T$ with respect to the standard basis $B=\{\mathbf{e}_1, \mathbf{e}_2\}$ of $\R^2$, where

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

1 \\

0

\end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}

0 \\

1

\end{bmatrix}.\]