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Lecture 8

Lecture 8 - Jan 31 2013 - Vector Spaces.pdf

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Department
Engineering
Course
ENGG 1500
Professor
Medhat Moussa
Semester
Winter

Description
Lecture 8 Chapter 4: Vector Spaces Vector Spaces  What is a vector space?  A set V together with an operation of addition, usually denoted x + y for any x, y  V , and an operation of scalar multiplication, usually denoted sx for any x  V and s  R, such that for any x, y, z  V and s, t,  R we have all of the following properties: (textbook, page 197) 2 V1 x + y  V (closed under addition) V2 (x + y) + z = x + (y + z) (addition is associative) V3 There is an element 0  V, called the zero vector, such that x + 0 = x = 0 + x (additive identity) V4 For each x  V there exists an element –x  V such that x + (-x) = 0 (additive inverse) V5 x + y = y + x (addition is commutative V6 sx  V (closed under scalar multiplication) V7 s(tx) = (st)x (scalar multiplication is associative) V8 (s + t) x = sx + tx (scalar addition is distributive) V9 s(x + y) = sx + sy (scalar multiplication is distributive) V10 1x = x (scalar multiplicative identity) 3 Example 1 Let P 2enote the set of all real polynomials of degree 2 or less. V erify that P2is a real vector space. 4 Solution Natural addition is associated with polynomials. For example, let p(x) and q(x) be the polynomials p(x) = 2x – x + 3 and q(x) = x + 2x – 1 Need to get the sum r(x) = p(x) + q(x) => r(x) = 3x + x + 2 5 Scalar multiplication: Let s(x) = 2q(x)  s(x) = 2x + 4x – 2  Seems reasonable to expect P t2 be a real vector space. Need to prove this more rigorously. 6 Define P to2be set of all expressions (or functions) of the form: p(x) = a x + a x + a 2 1 0 Where a , 2 , 1nd a are0real constants. 7 Define the following two vectors in P : 2 2 2 p(x) = a x2+ a x + 1 and q(0) = b x + b x + 2 1 0 r(x) = p(x) + q(x) = (a + b )x + (a + b )x + (a + b ) 2 2 1 1 0 0 Define the scalar multiple s(x) = cp(x) s(x) = (ca 2x + (ca )x1+ (ca ) 0 8 Subspaces (textbook, page 201)
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