8. Row Space, Column Space, Nullspace
8 Row space, column space, nullspace (kernel) 8.1 Subspaces The 2-dimensional plane, 3-dimensional space, Rn, Rm×n (m × n matrices) are all examples of Vector spaces. They all have the linear combination property: in any of these spaces, for any list X1,...,Xk of points from that space, and real numbers α1,...,αk, the space contains the linear combination α1X1 + ... + αkXk Other examples of vector subspace include • Any plane through O in 3 dimensions and similar... • We call the elements of a vector space points or vectors. • For any list X1,...,Xk of points, the set of all linear combinations α1X1 + ... + αkXk is a vector space in its own right. This set of all linear combinations is called the subspace spanned by the points X1,...,Xk. • For any m × n matrix A, there are three notable spaces associated with A: • the set of all linear combinations of rows of A, called the row space of A. • the set of all linear combinations of columns of A, called the column space of A. • The set of all column vectors X (in Rn×1) such that AX = O. This is called the kernel or nullspace of a. Row space, column space, and kernel (nullspace) of a matrix are all examples of ‘subspaces.’ The row space of Am×n is the space spanned by its rows and the column space is the space spanned by its columns. ‘Vectors’ usually mean column vectors. If we need to distinguish between row and column vectors, we write R1×n for the set of row vectors of width n, and Rm×1 for the column vectors of height m.
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