Basis

Definition

Let

• $V$ : Finite dimensional vector space
• $S=\{{\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_n\}$ : Set of vectors in $V$

If

• $S$ spans $V$
• $S$ is linearly independent

Then we say $S$ is basis for $V$

Remark

The difference between a $S$ being a basis or $V$ as opposed to simply spanning $V$ is the additional condition that $S$ is linearly independent, i.e., no vectors in $S$ can be expressed as a linear combination as other vectors in $S$.

In other words, Suppose $S$ spans $V$. If $S$ is linearly independent then $S$ is basis for $V$.

Finding basis for a vector space

Let

• $V$ : Finite dimensional vector space (e.g., $R^2$, $R^3$)
• $S=({\mathbf{v}}_1,{\mathbf{v}}_2,\dots ,{\mathbf{v}}_n)$ : Set of vectors in $V$
• $\mathbf{c}=(c_1,c_2,\dots ,c_n)$ : Coefficient

1. Prove $S$ is linearly independent
   • Show that the vector equation $S{\mathbf{c}}^T=c_1{\mathbf{v}}_1+c_2{\mathbf{v}}_2+\cdots +c_n{\mathbf{v}}_n=0$ has ONLY the trivial solution.
2. Prove $S$ spans $V$
   • Show that the every vector $\mathbf{b}=(b_1,b_2,\dots ,b_r)$ in $V$ can be expressed as $S{\mathbf{c}}^T=c_1{\mathbf{v}}_1+c_2{\mathbf{v}}_2+\cdots +c_n{\mathbf{v}}_n=\mathbf{b}$.

We can prove both point 1 and 2 by showing the matrix $A=S{\mathbf{c}}^T$ has a non 0 determinant ($\det (A)\ne 0$) by using statements (b), (e), and (g) of matrix equivalency statements theorem. That is,

• (b) $A\mathbf{x}=0$ has only the trivial solution
• (e) $A\mathbf{x}=\mathbf{b}$ is consistent for every $n\times 1$ matrix $\mathbf{b}$
• (g) $\det (A)\ne 0$

Example: Basis for $R^n$

Let

• ${\mathbf{v}}_1=(1,2,1)$
• ${\mathbf{v}}_2=(2,9,0)$
• ${\mathbf{v}}_3=(3,3,4)$

Following the procedure described in Finding basis for a vector space, our problem is now reduced into showing that the determinants of the following matrix is not 0:

$$\det \left(\left[\begin{array}{ccc}1& 2& 1\\ 2& 9& 0\\ 3& 3& 4\end{array}\right]\right)=-1\ne 0$$

Therefore, the set of vectors $S=\{{\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3\}$ is a basis for the vector space $R^3$.