notes6-1 - SECTION 6.1 INNER PRODUCT, LENGTH, ORTHOGONALITY...

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SECTION 6.1 INNER PRODUCT, LENGTH, ORTHOGONALITY SOME MOTIVATION. (1) Suppose we want to solve A x = b , but the system is incon- sistent. Then we’d like to do the best we can, that is, we’d like to find an x so that A x is as close as possible to b . We need to measure “closeness,” or distances in R p . (2) WE WANT OUR GEOMETRY BACK!!!! We want things like lengths and distances and angles and circles and . . . . We can’t expect to get geometry in abstract vector spaces, at least not right away, so we return to R n . It turns out that we get all the geometry we might want by means of the standard inner product or dot product of two vectors u = u 1 u 2 . . . u n and v = v 1 v 2 . . . v n in R n , which is defined to be u 1 v 1 + u 2 v 2 + ··· + u n v n . The inner product has natural nice algebraic properties listed on page 376. It gives rise to the length or norm of a vector u in R
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notes6-1 - SECTION 6.1 INNER PRODUCT, LENGTH, ORTHOGONALITY...

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