MATH 21 Lecture Notes - Lecture 20: Row And Column Spaces, Gaussian Elimination, Mexican Peso
Document Summary
Definition: nullity of mxn matrix a is dim (null (a)). Theorem: if matrices a & b are row equivalent then null (a) = null (b) Definition: column rank of a denoted by colrank (a). Dimension of column space of a (the span of columns) An immediate consequence of fact that row equivalent matrices have equal null spaces. Conclude: subsequence of columns of a is linearly dependent iff corresponding subsequence of columns of b is linearly independent. To extend to a basis of rn: make matrix a, use gaussian elimination. Definition: row rank of a, rowrank (a) is dimension of row space of a. Theorem: if a & b are row equivalent, row (a) = row (b) Theorem: the sequence of non-zero rows of a matrix in echelon form is linearly independent. Example: find basis of row space of a steps: find ref of a, any row with a pivot is the basis.
