Linear Independence
Chapter 1: Vector Space and Inner Product — Linear Independence (also appears in Ch. 2)
From the book
Chapter 1: Vector Space and Inner Product. In the chapter mind map this icon labels Linear Independence. The discussion below is excerpted and lightly edited from § Definition: Linear Independence in Mathematics for AI and Machine Learning. Related material also appears in Chapter 2 (Linear Independence & Matrix Rank $\mathrm{rank}(A)$).
Vectors $\mathbf{v}_0, \ldots, \mathbf{v}_{N-1} \in \mathbb{R}^d$ are linearly independent if the only solution $(c_0, \ldots, c_{N-1})$ to the equation $\sum_{n=0}^{N-1} c_n \mathbf{v}_n = \mathbf{0}$ is $c_0 = \cdots = c_{N-1} = 0$.
What this drawing shows
What you see. Depicts vectors pointing in nonredundant directions, indicating that no vector is a linear combination of the others.
In the mind map. Chapter 1 — Linear Independence. See From the book above for definitions, figures, and worked examples.
Also appears in Ch. 2 (Linear Independence & Matrix Rank).
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 Linear Independence.
This concept is also referenced in Chapter 2 (Linear Independence & Matrix Rank).
