Linear Algebra Chapter 4 Vector Spaces

4.1 General Vector Spaces Definition Our aim in this section will be to focus on the algebraic properties of Rn. 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 u + v…………… (V is closed under addition.) cu …………… (V is closed under scalar multiplication.)

Definition of Vector Space (continued) Addition Axioms 3. 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 Axioms 7. c(u + v) = …………… 8. (c + d)u = …………… c(du) = …………… 1u = ……………

A Vector Space in R3 Is V a vector space ? Is Z a vector space ? Example 1 Is V a vector space ? Solution Example 2 Is Z a vector space ? Solution Ch04_4

Prove that W is a vector space. Example 3 Prove that W is a vector space. Proof

Vector Spaces of Matrices (Mmn) Prove that M22 is a vector space. Proof

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

Example 4 Solution Ch04_8

Vector Spaces of Functions Prove that F = { f | f : R R } is a vector space.

Vector Spaces of Functions (continued)

Vector Spaces of Functions (continued) Example 5 Is the set F ={ f | f (x)=ax2+bx+c , a,b,c R , } a vector space? Solution Ch04_11

Subspaces Definition 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:

Example 6 Let U be the subset of R3 consisting of all vectors of the form (a, a, b) , a,bR , 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.

Example 7 Let 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

Example 8 Prove that the set W of 2  2 diagonal matrices is a subspace of the vector space M22. Solution Ch04_15

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