1 / 16

Chapter 4

Chapter 4. Vector Spaces. Linear Algebra. 4.1 General Vector Spaces. Our aim in this section will be to focus on the algebraic properties of R n. Definition

eben
Download Presentation

Chapter 4

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 4 Vector Spaces Linear Algebra

  2. 4.1 General Vector Spaces Our aim in this section will be to focus on the algebraic properties of Rn. • Definition • A ……………… is a set V of elements calledvectors, 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.)

  3. 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 = ……………

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

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

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

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

  8. Example 4 Solution Ch04_8

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

  10. Vector Spaces of Functions (continued)

  11. 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

  12. 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:

  13. 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.

  14. 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

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

  16. 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

More Related