Column Span
Chapter 1: Vector Space and Inner Product — Subspaces & Span (also appears in Ch. 2, Ch. 8)
From the book
Chapter 1: Vector Space and Inner Product. In the chapter mind map this icon labels Subspaces & Span. The discussion below is excerpted and lightly edited from § Definition: Linear Span in Mathematics for AI and Machine Learning. Related material also appears in Chapter 2 (Matrix-Vector (2 Views) & Matrix-Matrix (4 Views)), Chapter 8 (Least Squares Solution: $\mathbf{x}_{\mathrm{LS}} = A^\dagger\mathbf{b}$).
The linear span of vectors ${\mathbf{v}_0, \ldots, \mathbf{v}_{N-1}} \subset \mathbb{R}^d$ is defined as
For example, the span of $(1, 0, 0)^\top$ and $(0, 1, 0)^\top$ in $\mathbb{R}^3$ is the $xy$-plane $\{(a, b, 0)^\top : a, b \in \mathbb{R}\}$, and the span of a single nonzero vector is the line through the origin along it.
What this drawing shows
What you see. Shows columns as vectors whose linear combinations generate a subspace, emphasizing the range of a matrix.
In the mind map. Chapter 1 — Subspaces & Span. See From the book above for definitions, figures, and worked examples.
Also appears in Ch. 2 (Matrix-Vector (2 Views) & Matrix-Matrix (4 Views)); Ch. 8 (Least Squares Solution).
Where to read next
Read the full definitions, figures, and worked examples in Chapter 1: Vector Space and Inner Product — see the mind-map node Subspaces & Span.
This concept is also referenced in Chapter 2 (Matrix-Vector (2 Views) & Matrix-Matrix (4 Views)); Chapter 8 (Least Squares Solution).
