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Linear Algebra for Data Science Using Python

Span, Basis, and Dimensions

Explore the fundamental concepts of span, basis, and dimensions within vector spaces to build a strong foundation in linear algebra. Understand how spanning sets generate vector spaces, learn what constitutes a basis with linearly independent vectors, and grasp the fixed nature of dimensions across bases. This lesson equips you with essential knowledge for applying these concepts to data science and Python coding projects.

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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}

  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.

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 ...