Linear Algebra, 6 by Kenneth Hoffmann And Ray Kunze

By Kenneth Hoffmann And Ray Kunze

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Extra resources for Linear Algebra, 6

Example text

Ffy, row-equivalent to A. Then the row space of R is W. Now let R be any row-reduced echelon matrix which space. Let pl, . . , pr be the non-zero row vectors of R the leading non-zero entry of pi occurs in column ki, i vectors pl, . . , pI form a basis for W. In the proof of observed that if p = (b,, . . , b,) is in W, then with P = ClPl + * * * and ci = bk,; in other words, the unique bination of pl, . . , p, is (2-21) P = j, + of Row-Equitdence echelon matrix some m vectors matrix with row matrix which is has W as its row and suppose that = 1, .

A,} is an ordered basis for V. Now suppose V is a finite-dimensional vector and that 03 = (al, . . ) a,} is an ordered basis for V. Given (Xl, . * . , 2,) of scalars such that a! in V, there space over the field F is a unique n-tuple n Ck! = The n-tuple Z i=l X&ii. is unique, because if we also have then n 2 i=l (Xi - Xi)aCi = 0 and the linear independence of the ai tells us that xi - zi = 0 for each i. We shall call xi the ith coordinate of ar relative to the ordered basis 03 = {cdl,. ,Nn}.

Theorem 7. Let V be an n-dimensional vector space over the jield F, and let (a and 6~’ be two ordered bases of V. Then there is a unique, necessarily invertible, n X n matrix P with entries in F such that (ii; [~I63= P[~lW [a]@! = P-‘[ala for every vector o( in V. The columns of P are given by Pj = [41& j = 1, . . , n. Coordinates Sec. 4 To complete result. the above analysis we shall also prove the following Theorem 8. Suppose P is an n X n invertible matrix over F. Let V be an n-dimensional vector space over F, and let & be an ordered basis of V.

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