Basis Dimension

Chapter 1: Vector Space and Inner Product. In the chapter mind map this icon labels Basis & Dimension. The discussion below is excerpted and lightly edited from § Definition: Basis, Orthonormal Basis, Dimension of a Space in Mathematics for AI and Machine Learning. Related material also appears in Chapter 2 (Change of Basis: $B = P^{-1}AP$).

1. The vectors are linearly independent. 2. The vectors span $W$.

The dimension of $W$, denoted $\dim(W)$, is the number of vectors in a basis of $W$ (every basis has the same size, since an independent set is never larger than a spanning set).

Let $W$ be a space of dimension $r$. A set of vectors ${\mathbf{v}_0, \ldots, \mathbf{v}_{r-1}}$ is an orthonormal basis of space $W$ if

where $\delta_{ij}$ is the Kronecker delta (a scalar indicator function), defined as $\delta_{ij} = 1$ if $i = j$ and $\delta_{ij} = 0$ if $i \neq j$. This condition means that each vector has unit length ($\|\mathbf{v}_i\|_2 = 1$ for all $i$) and different vectors are orthogonal ($\langle \mathbf{v}_i, \mathbf{v}_j \rangle = 0$ for $i \neq j$).