Basis of Subspaces

A basis is a linearly independent set of vectors in a subspace $V$ that spans $V$ .

In simpler terms, if $V$ is a subspace, a collection of vectors in $V$ , ${v_{1}, v_{2}, \dots, v_{n}}$ , is a basis for $V$ if:

Theorem

Any vector $v_{1} \in V$ can be written in a unique way as a linear combination of elements of the basis.

Setup:

Proof:

Let’s make a homogeneous equation to prove linear independence.

0 = (c_{1} - d_{1}) v_{1} + (c_{2} - d_{2}) v_{2} + \dots + (c_{r} - d_{r}) v_{r}

if $c_{i} \neq = b_{i}$ then the basis in linearly dependent, which contradicts initial assumptions that $S$ is a basis.

Let $V$ be a vector space of polynomials of degree $\leq 2$

p_{1} = 1 - x, p_{2} = 1 + x, p_{3} = 2 + x^{2}

Then, ${p_{1}, p_{2}, p_{3}}$ is a basis for $V$ because:

The set $B$ is linearly independent.
The set $B$ spans $V$ because any polynomial of degree $\leq 2$ can be written as a linear combination of $p_{1}, p_{2}, p_{3}$ .