# mathc192 -4 - LECTURE NO.-4 3.1 VECTOR SPACES A (real)...

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A (real) vector space is a set V of objects called vectors , together with a rule for adding any two vectors u and v of V to produce a vector u +v in V and a rule for multiplying any vector u in V by any scalar α in R to produce a vector α u in V. LECTURE NO.-4 3.1 VECTOR SPACES

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Properties of Vector Addition Properties A1 through A4 and below are satisfied for all choices of vectors u, v, w V A1. (u+v) + w = u+(v+w) An associative law of addition A2 . u+v = v+u A commutative law A3. 0+u = u = 0 +u 0 as additive identity A4. u+(-u) =0 , -u as additive inverse of u
For all scalars α , β Properties Involving Scalar Multiplication S1 through S4 are satisfied: S1. α (u+v) = α u + α v scalar multiplication distributive over vector addition. S2. ( α + β ) u = α u + β u , A distributive law S3. α ( β u) = ( α β ) u = β ( α u), An associative law S4. 1u=u Existence of unit element 1.

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Example 1: Show that the set M m × n of all m × n matrices is a vector space, using the usual addition of matrices as vector addition and multiplication of a matrix by a scalar as scalar multiplication.
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## This note was uploaded on 05/14/2010 for the course MATHEMATIC mathe taught by Professor Xyz during the Spring '10 term at Birla Institute of Technology & Science.

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mathc192 -4 - LECTURE NO.-4 3.1 VECTOR SPACES A (real)...

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