Lecture4 - Lecture 4 4.1 Coordinatization of vectors Let B...

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1 Lecture 4 4.1 Coordinatization of vectors Let B = { 1 b , 2 b , …, n b } be an ordered basis for a finite-dimensional vector space V, and let n n b r b r b r v ... 2 2 1 1 . Then [ r 1 , r 2 , …, r n ] is the coordinate vector of v relative to the ordered basis B, and is denoted by B v . Example: Find the coordinate vector of the polynomial p(x) = x 2 5x 12 relative to the ordered basis B = {x 2 , x, 1}. The coordinate vector of v relative to an ordered basis of V is unique. In general, each p(x) = a n x n + a n 1 x n 1 + … + a 1 x + a 0 can be renamed by its coordinate vector [ a n , a n 1 , …, a 1 , a 0 ] relative to the ordered basis B = {x n , x n 1 , …, x , 1}. Coordinatization of a finite-dimensional vector space: V: vector space, B = n b b b ..., , , 2 1 V ------------------------ R n v B v w v B B B w v w v r v (r v ) B = r B v Whenever the vectors in a vector space V can be renamed to make appear structurally identical to a vector space W, we say V and W are isomorphic vector spaces . Every real vector space having a basis of n vectors is isomorphic to R n . Example: Determine whether S = {1, x + x 2 , 2x 3x 2 , 1 + 3x 2x 2 , x 3 } is independent.
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2 Example: Find the coordinate vector of [1, 2, 3] relative to the ordered basis
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