Question

In Exercises 19 and 20, $V$ is a vector space. Mark each statement True or False. Justify each answer.

19.

a. The number of pivot columns of a matrix equals the dimension of its column space.

b. A plane in ${\mathbb{R}}^3$ is a two-dimensional subspace of ${\mathbb{R}}^3$.

c. The dimension of the vector space ${\mathrm{P}}_4$ is 4.

d. If $\dim V=n$ and $S$ is a linearly independent set in $V$, then $S$ is a basis for $V$.

e. If a set $\left\{{\mathrm{v}}_1,...,{\mathrm{v}}_p\right\}$ spans a finite-dimensional vector space $V$ and if $T$ is a set of more than p vectors in $V$, then $T$ is linearly dependent.

Short answer

1. The given statement is true.
2. The given statement is false.
3. The given statement is false.
4. The given statement is false.
5. The given statement is true.

Step-by-step solution

Determine whether the given statement is true or false

a)

Thedimension of $\mathrm{Nul}A$ is thenumber of free variablesin the equation $A\mathrm{x}=0$, and the dimension of $\mathrm{Col}A$is thenumber of pivot columnsin $A$.

Thus, the given statement (a) is true.

Determine whether the given statement is true or false

b)

A plane in ${\mathbb{R}}^3$ is athree-dimensional subspace of ${\mathbb{R}}^3$.

Thus, the given statement (b) is false.

Determine whether the given statement is true or false

c)

Thestandard basisfor ${\mathbb{R}}^n$ contains $n$ vectors; so $\dim {\mathbb{R}}^n=n$. The standard polynomial basis $\left\{1,t,t^2\right\}$ shows that ${\mathrm{P}}_2=3$. In general, $\dim {\mathrm{P}}_n=n+1$.

Thus, the given statement (c) is false.

Determine whether the given statement is true or false

d)

Theorem 12states that let $V$ be a p-dimensional vector space; $p\ge 1$, then any linearly independent set of exactly $p$ elements in $V$is automatically a basis for $V$. Any set of exactly $p$ elements that span $V$is automatically a basis for $V$.

The set $S$ must have $n$ elements.

Thus, the given statement (d) is false.

Determine whether the given statement is true or false

e)

Theorem 9states that if a vector space $V$ has abasis $B=\left\{{\mathrm{b}}_1,...,{\mathrm{b}}_n\right\}$, then any set in $V$containing more than $n$ vectors must belinearly dependent.

Thus, the given statement (e) is true.