Page 511 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Proof: , then . _________
1. If x is a solution of

2. Thus, . _________

3. Thus, the column vectors of A are linearly independent. _________

Let A be an matrix with linearly independent column vectors, and let b be an

22. matrix. Give a formula in terms of A and for

(a) the vector in the column space of A that is closest to b relative to the Euclidean inner
product;

(b) the least squares solution of relative to the Euclidean inner product;

(c) the error in the least squares solution of relative to the Euclidean inner product;

(d) the standard matrix for the orthogonal projection of onto the column space of A
relative to the Euclidean inner product.

Refer to Exercises 18–20. Contrast the techniques of polynomial interpolation and fitting a line
23. by least squares. Give circumstances under which each is useful and appropriate.

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