Chapter 4 … part 1 4.1

Vector Spaces

Linear Algebra

s 8

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4.1 General Vector Spaces

Definition A ……………… is a set V of elements called vectors, having operations of addition and scalar multiplication defined on it that satisfy the following conditions:

Let u, v, and w be arbitrary elements of V, and c and d are scalars. • Closure Axioms 1. u + v…………… (V is closed under addition.)2. cu …………… (V is closed under scalar multiplication.)

Our aim in this section will be to focus on the algebraic properties of Rn.

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• Addition Axioms3. u + v = …………… (commutative property)4. u + (v + w) = …………… (associative property)5. There exists an element of V, called the …………, denoted 0,

such that u + 0 = ……6. called the ……… of u, such that u + (u) = 0.• Scalar Multiplication Axioms7. c(u + v) = ……………8. (c + d)u = ……………9. c(du) = …………… 10. 1u = ……………

Definition of Vector Space (continued)

u V u

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Example 1

, 3, 1, 1, 3, Let 5, 7, . { }V Is V a vector space ?

, 2, 1, 0, 1, 2, Let 3, 4, . { }Z Is Z a vector space ?

Solution

Solution

A Vector Space in R3

Example 2

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}. | 1) 0, (1, {Let R aaW Prove that W is a vector space.

Proof

Example 3

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Vector Spaces of Matrices (Mmn)

}.,,, | {Let 22 R

srqp

srqp

M Prove that M22 is a vector space.

Proof

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In general: The set of m n matrices, Mmn, is a vector space.

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{ | , , , 0} a vector space?p q

Is the set W p q r sr s

Example 4

Solution

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Vector Spaces of Functions

Prove that F = { f | f : R R } is a vector space.

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Vector Spaces of Functions (continued)

Is the set F ={ f | f (x)=ax2+bx+c , a,b,c R , } a vector space?0a

Example 5

Solution

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SubspacesDefinition Let V be a vector space and U be a …………………………. of V. U is said to be a …………… of V if it is ……………………….. and …………………………………..Note:

........

........

........ is a vector space

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Example 6Let U be the subset of R3 consisting of all vectors of the form (a, a, b) , a,bR , i.e., U = {(a, a, b) R3 }.Show that U is a subspace of R3.

Solution

Show that U = {(a, 0, 0) R3 , a R } is a subspace of R3.

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Example 7Let V be the set of vectors of of R3 of the form (a, a2, b), V = {(a, a2, b) R3 , a,b R }. Is V a subspace of R3 ?

Solution

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Example 8Prove that the set W of 2 2 diagonal matrices is a subspace of the vector space M22.

Solution

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Theorem 4.5 (Very important condition)Let U be a subspace of a vector space V. ………………………………………

Example 9 Let W be the set of vectors of the form (a, a, a+2). Show that W is not a subspace of R3.

Solution