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6 - Bases of Row Spaces and Null Spaces

6 - Bases of Row Spaces and Null Spaces - Math 1b Prac...

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Math 1b Prac — Bases for row spaces, null spaces; orthogonal spaces January 18, 2011 — slightly revised January 21, 2011 Given a matrix M , the row space of M is the span of its rows (the set of all linear combinations of its rows). The null space of M is the set of all column vectors x so that M x = 0 . In other words, the null space of M is the set of all vectors orthogonal to the rows of M . In yet other words, the null space of M is the “solution space” of the homogeneous system of linear equations corresponding to M . We will check (it is not hard) that these are really subspaces. Row operations on a matrix M do not change the row space or the nullspace of M . These notes are very concise. Proofs of the following theorems will be given in class, or possibly in an expanded version of this handout. We use the matrix M = 1 2 3 4 2 3 4 5 3 4 5 6 in our examples. Two ways to find a basis for the row space of A Theorem 1. Let E be a basic form of A . Then the nonzero rows of E form a basis for the row space of A . Example: The reduced echelon form of M is E = 1 0 1 2 0 1 2 3 0 0 0 0 . So one basis for the row space of M is, or consists of, (1 , 0 , 1 , 2) and (0 , 1 , 2 , 3). Theorem 2. Let E be a basic form of A . Then the the columns of A that correspond to the special columns of E
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