## Dimensions of General Vector Spaces

Definition

- The
**dimension**$\dim(V)$ of a vector space $V$ is the number of vectors in a basis for $V$.

Summary

Let $V$ be a vector space over a scalar field $K$. Suppose that \dim(V)=n$. Let $S=\{\mathbf{w}_1, \dots, \mathbf{w}_k\}$ be a set of vectors in $V$.

- The dimension of $V$ does not depend on the choice of a basis.
- If $W$ is a subspace of $V$, then $\dim(W)\leq \dim(V)$.
- If $k > n$, then the set $S$ is linearly dependent.
- If $k < n $, then the set $S$ cannot span the vector space $V$.

=solution

### Problems

- Let $V$ be the vector space of all $2\times 2$ matrices whose entries are real numbers. Let

\[W=\left\{\, A\in V \quad \middle | \quad A=\begin{bmatrix}

a & b\\

c& -a

\end{bmatrix} \text{ for any } a, b, c\in \R \,\right\}.\]**(a)**Show that $W$ is a subspace of $V$.

**(b)**Find a basis of $W$.

**(c)**Find the dimension of $W$.

*(The Ohio State University)* -
Let $V$ be the vector space of all $2\times 2$ matrices. Let $W$ be a subset of $V$ consisting of all $2\times 2$ skew-symmetric matrices. (Recall that a matrix $A$ is skew-symmetric if $A^{\trans}=-A$.)

**(a)**Prove that the subset $W$ is a subspace of $V$.

**(b)**Find the dimension of $W$.

(*The Ohio State University*) -
Let $P_4$ be the vector space consisting of all polynomials of degree $4$ or less with real number coefficients. Let $W$ be the subspace of $P_2$ by

\[W=\{ p(x)\in P_4 \mid p(1)+p(-1)=0 \text{ and } p(2)+p(-2)=0 \}.\] Find a basis of the subspace $W$ and determine the dimension of $W$. - Let $p_1(x), p_2(x), p_3(x), p_4(x)$ be (real) polynomials of degree at most $3$. Which (if any) of the following two conditions is sufficient for the conclusion that these polynomials are linearly dependent?

**(a)**At $1$ each of the polynomials has the value $0$. Namely $p_i(1)=0$ for $i=1,2,3,4$.

**(b)**At $0$ each of the polynomials has the value $1$. Namely $p_i(0)=1$ for $i=1,2,3,4$.

(*University of California, Berkeley*) -
Let $V$ be the vector space of all $3\times 3$ real matrices. Let $A$ be the matrix given below and we define

\[W=\{M\in V \mid AM=MA\}.\] That is, $W$ consists of matrices that commute with $A$. Then $W$ is a subspace of $V$. Determine which matrices are in the subspace $W$ and find the dimension of $W$.

**(a)**\[A=\begin{bmatrix}

a & 0 & 0 \\

0 &b &0 \\

0 & 0 & c

\end{bmatrix},\] where $a, b, c$ are distinct real numbers.

**(b)**\[A=\begin{bmatrix}

a & 0 & 0 \\

0 &a &0 \\

0 & 0 & b

\end{bmatrix},\] where $a, b$ are distinct real numbers. - Let $U$ and $V$ be finite dimensional subspaces in a vector space over a scalar field $K$. Then prove that $\dim(U+V) \leq \dim(U)+\dim(V)$.
- Let $A$ and $B$ be $m\times n$ matrices. Prove that $\rk(A+B) \leq \rk(A)+\rk(B)$.