C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes6-1 -...

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SECTION 6.1 INNER PRODUCT, LENGTH, ORTHOGONALITY The 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 is 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 n given by || u || = u · u . This agrees (at least in R 2 ) with our understanding of the length of the line segment from the origin to the tip of u . A unit vector is a vector whose length is 1. If we divide a nonzero vector by its length, we’ll get a unit vector in the same direction. The distance between vectors u and v in R n is dist( u , v ) = || u - v || . Again, this agrees with the geometry, at least in R 2 . THE GEOMETRY IN R 2 . EXAMPLE. For x = 3 - 2 5 and w = - 1 2 1 , compute x · w , || x || , and a unit vector in the direction of x .
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From the Law of Cosines we find that u · v = || u |||| v || cos θ , where θ is the angle between
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Unformatted text preview: the two vectors. This implies that two nonzero vectors are perpendicular or orthogonal exactly when their dot product is 0. THE PYTHAGOREAN THEOREM. Two vectors u and v are orthogonal exactly when || u + v || 2 = || u || 2 + || v || 2 . DRAW A PICTURE AND CALCULATE. If W is a subspace of R n and if the vector x in R n is orthogonal to every vector in W , then we say that x is orthogonal to W . The set of all vectors orthogonal to W is called the orthogonal complement of W and is denoted by W . FACTS ABOUT ORTHOGONAL COMPLEMENTS. (1) A vector x is in W ex-actly when x is orthogonal to every vector in a set that spans W . (2) W is a subspace of R n . (3) (Row A ) = Nul A and (Col A ) = Nul A T . HOMEWORK: SECTION 6.1...
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This note was uploaded on 04/15/2008 for the course M 340L taught by Professor Pavlovic during the Spring '08 term at University of Texas at Austin.

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C__DOCUME~1_MAXWID~1_LOCALS~1_Temp_plugtmp-27_notes6-1 -...

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