4.4_4.5 - MATH1850U/2050U Chapter 4 cont 1 GENERAL VECTOR...

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MATH1850U/2050U: Chapter 4 cont. .. 1 GENERAL VECTOR SPACES cont. .. Coordinates and Basis (4.4; pg. 200) Note regarding non-rectangular coordinate systems: A coordinate system establishes a one-to-one correspondence with all points in the plane, and we use an ordered pair to do this. The numbers in the ordered pair usually correspond to units along perpendicular axis, but this is not necessary. Any two nonparallel lines can be used to define a coordinate system in the plane. Note: We can specify a coordinate system in terms of basis vectors. Note: For a coordinate system, we need: Unique representation of everything in the vector space Ability to “label” everything in the vector space Note: A basis can be used to define a coordinate system in a vector space. Example: n e e e , , , 2 1 are a basis for R n . Definition: If V is any vector space and   n S v v v , , , 2 1 is a set of vectors in V , then S is called a basis for V if the following two conditions hold: a) S is linearly independent b) S spans V
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MATH1850U/2050U: Chapter 4 cont. .. 2 Theorem (Uniqueness of Basis Representation): If   n S v v v , , , 2 1 is a basis for a vector space V , then every vector v in V can be expressed in the form n n c c c v v v v 2
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4.4_4.5 - MATH1850U/2050U Chapter 4 cont 1 GENERAL VECTOR...

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