Linear Algebra Chapter 4 Part 2 Vector Spaces

Linear Algebra Chapter 4 - Part 2 Vector Spaces

4. 2 Linear Combinations of Vectors Definition Let v 1, v 2, …, vm be vectors in a vector space V. We say that , is a …………… of ……………… , if there exist ……………. . … such that v can be write as ……………. . … Example 1 The vector (7, 3, 2) is a linear combination of the vectors (1, 3, 0), (2, -3, 1) since: Solution Ch 04_2

Example 2 Determine whether or not the vector (8, 0, 5) is a linear combination of (1, 2, 3), (0, 1, 4), and (2, -1, 1). Solution Ch 04_3

Example 3 Can the vector (3, -4, -6) be a linear combination of (1, 2, 3), (-1, -2), and (1, 4, 5). Solution Ch 04_4

Example 4 Express the vector (4, 5, 5) as a linear combination of the vectors (1, 2, 3), (-1, 1, 4), and (3, 3, 2). Solution Ch 04_5

Example 5 Determine whether the matrix of the matrices is a linear combination in the vector space M 22 Solution Ch 04_6

Example 6 Determine whether the function combination of the functions is a linear and Solution Ch 04_7

Spanning Sets Definition The vectors v 1, v 2, …, vm are said to …………… a vector space if every vector in the space can be expressed as a ……………. of these vectors. Ch 04_8

Example 7 Show that the vectors (1, 2, 0), (0, 1, -1), and (1, 1, 2) span R 3. Solution Ch 04_9

Example 8 Show that the following matrices span the vector space M 22 of 2 2 matrices. Solution Ch 04_10

4. 3 Linear Dependence and Independence Definition (a) The set of vectors { v 1, …, vm } in a vector space V is said to be …………. . …… if there exist scalars c 1, …, cm …………, such that ………………. …… (b) The set of vectors { v 1, …, vm } is ………………. . … if ……. . ………… can only be satisfied when ……………… Ch 04_11

Example 9 Show that the set {(1, 2, 0), (0, 1, -1), (1, 1, 2)} is linearly independent in R 3. Solution Ch 04_12

Example 10 Show that the set of functions {x + 1, x – 1, – x + 5} is linearly dependent in P 1. Solution Ch 04_13

Theorem 4. 7 A set consisting of two or more vectors in a vector space is ………………… it is possible to express ……… of them as a …………………… Example 11 The set of vectors {(1, 2, 1) , (-1, 0) , (0, 1, 1)} is linearly ………………………. . … Example 12 The set of vectors {(2, -1, 3) , (4, -2, 6)} is linearly ………………………. . … Example 13 The set of vectors {(1, 2, 3) , (6, 5, 4)} is linearly ………………………. . … Ch 04_14

Theorem 4. 8 Let V be a vector space. Any set of vectors in V that contains the……. is linearly …………. Example 14 The set of vectors {0, v 1, v 2, … , vn} is linearly………………… Ch 04_15

Theorem 4. 9 Let the set {v 1, …, vm} be linearly …………. . . in a vector space V. Any set of vectors in V ……………. will …… be linearly ………………. Example 15 W={(1, 2, 3) , (2, 4, 6)} is linearly …………… U={(1, 2, 3) , (2, 4, 6), (4, 5, 6), (3, 5, 4)} is ………… Note Let the set {v 1, v 2} be linearly independent, then {v 1 + v 2, v 1 – v 2} is also linearly…………… Ch 04_16

4. 4 Bases and Dimension Definition A finite set of vectors {v 1, …, vm} is called a …………for a vector space V if: 1. the set ………………. . 2. the set ………………. . Standard Basis The set of n vectors ……………………… is a ……. . for Rn. This basis is called the …………. basis for Rn. Theorem 4. 11 Any two bases for a vector space V consist of the ………………. . Ch 04_17

Definition If a vector space V has a basis consisting of n vectors, then the ……………. . of V is said to be n and denoted by …………. . . Note Ch 04_18

Example 16 Prove that the set {(1, 0, -1), (1, 1, 1), (1, 2, 4)} is a basis for R 3. Solution Ch 04_19

Example 17 Show that { f, g, h }, where f(x) = x 2 + 1, g(x) = 3 x – 1, and h(x) = – 4 x + 1 is a basis for P 2. Solution Ch 04_20

Theorem 4. 10 Let {v 1, …, vn } be a basis for a vector space V. If {w 1, …, wm} is a set of …………… vectors in V, then this set is linearly ………………. Example 18 Is the set {(1, 2), (-1, 3), (5, 2)} linearly independent in R 2. Solution Ch 04_21

Theorem 4. 14 Let V be a vector space of dim(V)= n. (a) If S = {v 1, …, vn} is a set of n …………. vectors in V S is a ………. for V. (b) If S = {v 1, …, vn} is a set of n vectors V that …………… V S is a ………. for V. Ch 04_22

Example 19 Prove that the set B={(1, 3, -1), (2, 1, 0), (4, 2, 1)} is a basis for R 3. Solution Ch 04_23

Example 20 State (with a brief explanation) whether the following statements are true or false. (a) The vectors (1, 2), (-1, 3), (5, 2) are linearly dependent in R 2. (b) The vectors (1, 0, 0), (0, 2, 0), (1, 2, 0) span R 3. (c) {(1, 0, 2), (0, 1, -3)} is a basis for the subspace of R 3 consisting of vectors of the form (a, b, 2 a -3 b). (d) Any set of two vectors can be used to generate a twodimensional subspace of R 3. Note (b) dim(V)=n: *{v 1, …, vn}Span V then it is linearly independent *Not linearly independent then not span. Solution Ch 04_24