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Lecture 2

Week 2 (lec 4-6) Lecture Notes

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University of Waterloo
MATH 136
Patrick Roh

Lec 4 SubspaceDot ProductWednesday January 11 2012926 AMRecall Iffor anythenis closed under addition Note This is property 1 of Iffor any andthenis closed under scalar multiplication Property 6Definition If a nonempty subsetofsatisfies the ten properties ofthen it is a subspace ofTheorem Subspace Test Letbe a nonempty subset of Ifandfor alland thenis a subspace ofProof Properties 2 3 7 8 9 10 must hold because we are applying the same operations as inProperties 1 and 6 are what is being checked Properties 4 holds sincefor any Properties 5 holds since for anyNote Our proof shows that any set that does NOT contain the zero vector is NOT a subspaceEgis NOT a subspace because is a plane through the point Eg Note sois nonemptyLetLet Note 0Property 4 holds true MATH 136 Page 1NoteBy the subspace testis a subspace ofNotea line through the origin with direction vectorEg Note sois nonemptyLet Butis NOT a subspaceDot ProductDefinition Given vectorsthe dot productof and is The dot product is also referred to as the standard inner productor the scalar productwhen inNote The result of the dot product is a scalar NOT a vectorEgMATH 136 Page 2
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