MATH125 Lecture Notes - Lecture 21: Linear Combination, Row And Column Spaces

• School: University of Alberta
• Department: Mathematics
• Course: MATH125
• Professor: George Peschke

Document Summary

A basis of subvector space v of ^n is a collection of vectors b from v satisfying: (i) b spans v (ii) b is linearly independant. The vectors s = {e1, , en} form a basis of ^n; the standard basis of ^n (1, 0, , 0) (0, 1, , 0) (0, , 1, 0) (0, , 0, 1) Suppose b = {b1, bk} is a basis of the subspace v of ^n. Then every x v has uniquely determined numbers x1, x2, , xk such that x = x1b1 + + xkbk. Why? (i) span(b) = v means, each x in v satisfies x = s1b1 + + skbk (ii) b linear independance means: there is at most one such set of numbers. So, every vector x in v can be expressed as a linear combination of the vectors in b x = x1b1 + + xkbk; for some numbers x1, , xn.