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The column space of a matrix is a vector space

Created over 8 years ago, updated 10 days ago

Theorem: The column space of a matrix (A) (the span of its column vectors) is a vector space --- specifically, a subspace of (\mathbb{F}^m) where (m) is the number of rows.

Notation: ( ext{col}(A)), ( ext{colspace}(A)), or ( ext{im}(A)).

The column space equals the range of the linear transformation (T(\mathbf{x}) = A\mathbf{x}).

Key fact: The dimension of the column space is the rank of (A). Row rank equals column rank.

Example: For (A = egin{pmatrix} 1 & 2 \ 3 & 6 \ 0 & 0 \end{pmatrix}), the column space is spanned by (egin{pmatrix} 1 \ 3 \ 0 \end{pmatrix}) (since column 2 = 2 times column 1), a 1-dimensional subspace of (\mathbb{R}^3).