by Seth Grief-Albert

The basis of a vector space is a set of linearly independent vectors that span the full space.

R2

$\R^2$ contains two important vectors: $\hat i$ and $\hat j$: The basis of a coordinate system.

Basis

A basis is a spanning set that is linearly independent. It contains the minimum number of vectors to reach an entire space.

Spanning Set

A spanning {or generating} set can spread its arms out to reach the entire space, regardless of independence.

$$ S{\{v_1,...,v_p\}}=V $$

Proof of a Generating Set

$$ v=\begin{bmatrix} a\\b\\c \end{bmatrix}\&\hspace{.1cm}v_1= \begin{bmatrix} 1\\0\\0 \end{bmatrix}\&\hspace{.1cm} v_2=\begin{bmatrix} 0\\1\\0 \end{bmatrix}\&\hspace{.1cm}v_3= \begin{bmatrix} 0\\0\\1 \end{bmatrix} $$

If the vectors represent a spanning set, v can be constructed as a linear combination with a solution. Thus, to prove that a set is spanning is to show that any target vector in the space is accessible (surjective😎)

$$ a \begin{bmatrix} 1\\0\\0 \end{bmatrix}+b \begin{bmatrix} 0\\1\\0 \end{bmatrix}+ c\begin{bmatrix} 0\\0\\1 \end{bmatrix} $$

<aside> 💡 Note: in a p-tuple, order matters.

</aside>

Coordinates With Respect to a Basis

Coordinates of a given vector can be represented in terms (linear combinations) of the n basis vectors

Dimensionality

The dimension of V is the number of vectors in a basis for V. If V doesn’t have a finite basis then dim(V) is infinite.

$$ R^n\to n\hspace{.1cm}dimensions\\ P_n\to n+1\hspace{.1cm}dimensions\\ C^\infin\to\infin \hspace{.1cm}dimensions $$

To compute the dimension of a vector space is to find a basis for that vector space.