Geometry and algebra of linear maps, vectors, and matrices
Three quantities capture what a matrix really does. The column space is everything Ax can reach: the span of the columns, the matrix's "output region." The rank is the dimension of that column space, the number of genuinely independent directions A produces. And the null space is everything A crushes to zero, all x with Ax = 0.
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▶ Rank, Nullspace, Column Space