Chapter 4 - Part 2

Vector Spaces

Linear Algebra

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4.2 Linear Combinations of Vectors

Definition Let v1, v2, …, vm be vectors in a vector space V.

We say that , is a ……………………… of

……………… , if there exist ……………..… such that

v can be write as ………………………..…

v V

Example 1

Solution

The vector (7, 3, 2) is a linear combination of the vectors

(1, 3, 0), (2, -3, 1) since:

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Example 2

Determine whether or not the vector (, 0, 5) is a linear combination of (1, 2, 3), (0, 1, 4), and (2, -1, 1).

Solution

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Example 3

Can the vector (3, 4, 6) be a linear combination of (1, 2, 3), (1, 1, 2), and (1, 4, 5).

Solution

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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

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Example 5

Determine whether the matrix is a linear combination

of the matrices in the vector space M22

1871

0210

and ,2032

,1201

Solution

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Example 6Determine whether the function is a linear combination of the functions and

710)( 2 xxxf

Solution

13)( 2 xxxg

.42)( 2 xxxh

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Definition The vectors v1, v2, …, vm are said to …………… a vector space if every vector in the space can be expressed as a …………………………. of these vectors.

Spanning Sets

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Show that the vectors (1, 2, 0), (0, 1, 1), and (1, 1, 2) span R3.

Solution

Example 7

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Example 8Show that the following matrices span the vector space M22 of 2 2 matrices.

1000

0100

0010

0001

Solution

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4.3 Linear Dependence and Independence

Definition(a) The set of vectors { v1, …, vm } in a vector space V is said to

be …………..…… if there exist scalars c1, …, cm …………, such that ……………….……

(b) The set of vectors { v1, …, vm } is ………………..… if ……..………… can only be satisfied when ………………

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Example 9Show that the set {(1, 2, 0), (0, 1, -1), (1, 1, 2)} is linearly independent in R3.

Solution

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Example 10Show that the set of functions {x + 1, x – 1, – x + 5} is linearly dependent in P1.

Solution

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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, -1, 0) , (0, 1,1)} is linearly ………………………………………………..…

m

The set of vectors {(2, -1, 3) , (4, -2, 6)} is linearly ………………………………………………..…

The set of vectors {(1, 2, 3) , (6, 5, 4)} is linearly ………………………………………………..…

Example 12

Example 13

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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,v1, v2, … , vn} is linearly……………………………………

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Theorem 4.9Let the set {v1, …, 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 ……………………

Let the set {v1, v2} be linearly independent, then {v1 + v2, v1 – v2} is also linearly………………………

Note

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4.4 Bases and Dimension

Definition

A finite set of vectors {v1, …, 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 ………………..

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DefinitionIf a vector space V has a basis consisting of n vectors, then the …………….. of V is said to be n and denoted by …………...

Note

dim ........

dim ........

dim ..........

n

mn

n

R

M

P

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Example 16

Prove that the set {(1, 0, 1), (1, 1, 1), (1, 2, 4)} is a basis for R3.Solution

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Example 17

Show that { f, g, h }, where f(x) = x2 + 1, g(x) = 3x – 1, and h(x) = –4x + 1 is a basis for P2.

Solution

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Theorem 4.10 Let {v1, …, vn } be a basis for a vector space V. If {w1, …, wm} is a set of …………… vectors in V, then this set is linearly ……………….

Example 18

Solution

Is the set {(1, 2), (-1, 3), (5, 2)} linearly independent in R2.

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Theorem 4.14

Let V be a vector space of dim(V)= n.

(a) If S = {v1, …, vn} is a set of n ……………………. vectors in V S is a ………. for V.

(b) If S = {v1, …, vn} is a set of n vectors V that …………… V S is a ………. for V.

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Example 19Prove that the set B={(1, 3, -1), (2, 1, 0), (4, 2, 1)} is a basis for R3.

Solution

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Example 20State (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 R2.

(b) The vectors (1, 0, 0), (0, 2, 0), (1, 2, 0) span R3.

(c) {(1, 0, 2), (0, 1, -3)} is a basis for the subspace of R3 consisting of vectors of the form (a, b, 2a 3b).

(d) Any set of two vectors can be used to generate a two-dimensional subspace of R3.

dim(V)=n: *{v1, …, vn}Span V then it is linearly independent

*Not linearly independent then not span.

Note (b)

Solution