{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

6 - Bases of Row Spaces and Null Spaces

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}