# Lecture2 - Lecture 2 2.1 General vector space (Cont.) In...

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1 Lecture 2 2.1 General vector space (Cont.) In fact, the set of all polynomials P ( F ) with coefficients from a field F is a vector space. Example: Let the set R 2 with addition: (u 1 , u 2 ) (v 1 , v 2 ) = (u 1 + v 1 ,u 2 + v 2 + 1) scalar multiplication: a (u 1 , u 2 ) = ( a u 1 , a u 2 + a – 1). Is the set over R a real vector space?

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2 Example: Properties: Let V be a General vector space over F. 1. 0 is the unique vector x satisfying x v = v , for v V. 2. v V, - v is the unique vector y satisfying y v = 0 . 3. If u v = u w , w v u , , V, then v = w . 4. 0 v = 0 , for v V. 5. r 0 = 0 , r F. 6. (- r) v = r (- v ) = - (r v ), for r F, v V. .
3 Example: Is the set V of all polynomials of degree 3, together with usual operations of P vector space? If not, why not? Question: How about The set V of all 2 2 matrices of the form c b 0 0 ; usual operations of matrices.

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## This note was uploaded on 12/25/2010 for the course MATHEMATIC MATB24 taught by Professor Yang during the Fall '09 term at University of Toronto- Toronto.

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Lecture2 - Lecture 2 2.1 General vector space (Cont.) In...

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