lecture_31 - MA 265 LECTURE NOTES: FRIDAY, MARCH 28 Inner...

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MA 265 LECTURE NOTES: FRIDAY, MARCH 28 Inner Product Spaces Length and Direction in Inner Product Spaces. Let ( V, + , · , ( , ) ) be an inner product space. We define length, distance, and direction in the following way: The length of a vector v V is given by || v || = p ( v , v ). The distance between two vectors u , v V is given by d ( u , v ) = || v - u || = p ( v - u , v - u ) = p || u || 2 - 2 ( u , v ) + || v || 2 . The angle between two nonzero vectors u , v V is given by θ = arccos ( u , v ) || u |||| v || where 0 θ π. By (Positivity), we see that || v || = 0 if and only if v = 0 . By (Linearity), we see that || c v || = | c ||| v || for any scalar c R . Using these definitions, we say that two vectors u and v – not necessarily nonzero! – are in the same direction if ( u , v ) = || u || || v || ; orthogonal or perpendicular if ( u , v ) = 0; and in opposite directions if ( u , v ) = -|| u || || v || . Orthonormal Sets. Let ( V, + , · , ( , ) ) be an inner product space. Say that we have a collection S = { v 1 , v 2 , ..., v n } of vectors from V . We say that S is orthogonal if we have ( v i , v j ) = 0 for all i 6 = j . We say that a vector v is normal if each it has length 1 i.e., || v || = 1. We use the portmanteau orthonormal if S is orthogonal and each vector in S is normal. Let
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lecture_31 - MA 265 LECTURE NOTES: FRIDAY, MARCH 28 Inner...

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