Linear Algebra with Applications, 10th Edition

3.4 Basis and Dimension

In Section 3.3, we showed that a spanning set for a vector space is minimal if its elements are linearly independent. The elements of a minimal spanning set form the basic building blocks for the whole vector space, and consequently, we say that they form a basis for the vector space.

Definition

The vectors ${\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_n$ form a basis for a vector space V if and only if

1. ${\mathbf{v}}_1,\dots ,{\mathbf{v}}_n$ are linearly independent.

2. ${\mathbf{v}}_1,\dots ,{\mathbf{v}}_n$ span V.

Example 1

The standard basis for ${ℝ}^3$ is $\{{\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3\}$; however, there are many bases that we could choose for ${ℝ}^3$. For example,

$$\begin{array}{ccc}\{\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right],\left[\begin{array}{c}0\\ 1\\ 1\end{array}\right],\left[\begin{array}{c}2\\ 0\\ 1\end{array}\right]\}& \text{and}& \{\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right],\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right],\left[\begin{array}{c}1\\ 0\\ 1\end{array}\right]\}\end{array}$$

are both bases for ${ℝ}^3$. We will see shortly that any basis for ${ℝ}^3$ must have exactly three elements.

Example 2

In ${ℝ}^{2\times 2}$, consider ...