chapter 4 - part 2
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Chapter 4 - Part 2
Vector Spaces
Linear Algebra
Ch04_2
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:
Ch04_3
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
Ch04_4
Example 3
Can the vector (3, 4, 6) be a linear combination of (1, 2, 3), (1, 1, 2), and (1, 4, 5).
Solution
Ch04_5
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
Ch04_7
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
Ch04_9
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 ………………
Ch04_12
Example 9Show that the set {(1, 2, 0), (0, 1, -1), (1, 1, 2)} is linearly independent in R3.
Solution
Ch04_13
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
Ch04_20
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
Ch04_21
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
Ch04_24
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
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