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Span, Basis, and Dimensions

Span, Basis, and Dimensions

Learn the concepts of the span, the dimension of a vector space, and the basis of a vector space.

Span

The span of a set, UU, is another set, VV, consisting of all the linear combinations of vectors in UU. The set UU is referred to as a spanning set.

Examples

  1. span({[10],[01]})=R2span\bigg(\bigg\{\begin{bmatrix}1\\0\end{bmatrix},\begin{bmatrix}0\\1\end{bmatrix}\bigg\}\bigg)=\R^2

We can create any vector in R2\R^2, say [αβ]T\begin{bmatrix}\alpha & \beta\end{bmatrix}^T, using a linear combination of the spanning set X={[10]T,[10]T}X=\bigg\{\begin{bmatrix}1 & 0\end{bmatrix}^T, \begin{bmatrix}1 & 0\end{bmatrix}^T\bigg\}. So, XX spans R2\R^2. That is:

α[10]+β[01]=[αβ]\alpha\begin{bmatrix}1 \\ 0\end{bmatrix}+\beta\begin{bmatrix}0 \\ 1\end{bmatrix}=\begin{bmatrix}\alpha \\ \beta\end{bmatrix}

Linear combinations of spanning set
  1. span({[ab]})=span\bigg(\bigg\{\begin{bmatrix}a\\b\end{bmatrix}\bigg\}\bigg)= line in the direction of [ab]\begin{bmatrix}a\\b\end{bmatrix}

Note: A line consists of points that are also position-vectors.

Different scalings of the vector generate all the points of a line

Spanning set isn’t unique

A spanning set corresponding to a given span isn’t unique.

In a previous example, we showed that span({[10],[01]})=R2span\bigg(\bigg\{\begin{bmatrix}1\\0\end{bmatrix},\begin{bmatrix}0\\1\end{bmatrix}\bigg\}\bigg)=\R^2 . . However, there exists infinitely many spanning sets that span R2\R^2 ...