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Span, Linear
Independence,
Dimension
Math 240
Spanning sets
Linear
independence
Bases andDimension
Span, Linear Independence, and Dimension
Math 240 — Calculus III
Summer 2013, Session II
Thursday, July 18, 2013
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Span, Linear
Independence,
Dimension
Math 240
Spanning sets
Linear
independence
Bases andDimension
Agenda
1. Spanning sets
2. Linear independence
3. Bases and Dimension
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Span, Linear
Independence,
Dimension
Math 240
Spanning sets
Linear
independence
Bases andDimension
Recap of span
Yesterday, we saw how to construct a subspace of a vectorspace as the span of a collection of vectors.
QuestionWhat’s the span of v1 = (1, 1) and v2 = (2,−1) in R
2?
Answer: R2.
Today we ask, when is this subspace equal to the whole vector
space?
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Span, Linear
Independence,
Dimension
Math 240
Spanning sets
Linear
independence
Bases andDimension
Definition
DefinitionLet V be a vector space and v1, . . . , vn ∈ V . The set{v1, . . . , vn} is a spanning set for V if
span{v1, . . . , vn} = V.
We also say that V is generated or spanned by v1, . . . , vn.
TheoremLet v1, . . . , vn be vectors in R
n. Then {v1, . . . , vn} spans Rn
if and only if, for the matrix A =v1 v2 · · · vn
, the
linear system Ax = v is consistent for every v ∈ Rn.
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Span, Linear
Independence,
Dimension
Math 240
Spanning sets
Linear
independence
Bases andDimension
Example
Determine whether the vectors v1 = (1,−1, 4),
v2 = (−2, 1, 3), and v3 = (4,−3, 5) span R3.
Our aim is to solve the linear system Ax = v, where
A =
1 −2 4
−1 1 −34 3 5
and x
=
c1
c2c3
,
for an arbitrary v ∈ R3. If v = (x,y,z), reduce the augmented
matrix to
1 −2 4 x
0 1 −1 −x − y
0 0 0 7x + 11y + z
.
This has a solution only when 7x + 11y + z = 0. Thus, thespan of these three vectors is a plane; they do not span R
3.
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Span, Linear
Independence,
Dimension
Math 240
Spanning sets
Linear
independence
Bases andDimension
Definition
Definition
A set of vectors {v1, . . . , vn} is said to be linearly dependentif there are scalars c1, . . . , cn, not all zero , such that
c1v1 + c2v2 + · · · + cnvn = 0.
Such a linear combination is called a linear dependence
relation or a linear dependency. The set of vectors is linearlyindependent if the only linear combination producing 0 is thetrivial one with c1 = · · · = cn = 0.
Example
Consider a set consisting of a single vector v. If v = 0 then {v} is linearly dependent because, for
example, 1v = 0.
If v = 0 then the only scalar c such that cv = 0 is c = 0.
Hence, {v
} is linearly independent.
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Span, Linear
Independence,
Dimension
Math 240
Spanning sets
Linear
independence
Bases andDimension
Practice
1. Find a linear dependency among the vectors
f 1(x) = 1, f 2(x) = 2 sin2 x, f 3(x) = −5cos2 x
in the vector space C 0(R).
2. If v1 = (1, 2,−1), v2 = (2,−1, 1), and v3 = (8, 1, 1),
show that {v1, v2,v3} is linearly dependent in R3 byexhibiting a linear dependency.
Proposition
Any set of vectors that are not all zero contains a linearly
independent subset with the same span.
Proof.Remove 0 and any vectors that are linear combinations of theothers. Q.E .D.
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Span, Linear
Independence,
Dimension
Math 240
Spanning sets
Linear
independence
Bases andDimension
Linear independence of functions
Definition
A set of functions {f 1, f 2, . . . , f n} is linearly independent onan interval I if the only values of the scalars c1, c2, . . . , cnsuch that
c1f 1(x) + c2f 2(x) + · · · + cnf n(x) = 0 for all x ∈ I
are c1 = c2 = · · · = cn = 0.
DefinitionLet f 1, f 2, . . . , f n ∈ C n−1(I ). The Wronskian of thesefunctions is
W [f 1, . . . , f n](x) =
f 1(x) f 2(x) · · · f n(x)f 1(x) f 2(x) · · · f n(x)
... ...
. . . ...
f (n−1)1 (x) f
(n−1)2 (x) · · · f
(n−1)n (x)
.
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Span, Linear
Independence,
Dimension
Math 240
Spanning sets
Linear
independence
Bases andDimension
Linear independence of functions
TheoremLet f 1, f 2, . . . , f n ∈ C n−1(I ). If W [f 1, f 2, . . . , f n] is nonzero at
some point in I then {f 1, . . . , f n} is linearly independent on I .
Remarks
1. In order for {f 1, . . . , f n} to be linearly independent on I ,it is enough for W [f 1, . . . , f n] to be nonzero at a singlepoint.
2. The theorem does not say that the set is linearlydependent if W [f 1, . . . , f n](x) = 0 for all x ∈ I .
3. The Wronskian will be more useful in the case wheref 1, . . . , f n are the solutions to a differential equation, inwhich case it will completely determine their lineardependence or independence.
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Span, Linear
Independence,
Dimension
Math 240
Spanning sets
Linear
independence
Bases andDimension
Minimal spanning sets
Since we can remove vectors from a linearly dependent set
without changing the span, a “minimal spanning set” should belinearly independent.
DefinitionA set of vectors {v1,v2, . . . , vn} in a vector space V is called abasis (plural bases) for V if
1. The vectors are linearly independent.
2. They span V .
Examples
1. The standard basis for Rn is
e1 = (1, 0, 0, . . . ), e2 = (0, 1, 0, . . . ), . . .
2. Any linearly independent set is a basis for its span.
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Span, Linear
Independence,
Dimension
Math 240
Spanning sets
Linearindependence
Bases andDimension
Dimension
R3 has a basis with 3 vectors. Could any basis have more?
Suppose v1,v2, . . . , vn is another basis for R3 and n > 3.Express each v j as
vi = (v1 j , v2 j , v3 j) = v1 je1 + v2 je2 + v3 je3.
If A =
v1 v2 · · · vn
= [vij ]
then the system Ax = 0 has a nontrivial solution becauserank(A) ≤ 3. Such a nontrivial solution is a linear dependency
among v
1,v2, . . . ,
vn, so in fact they do not form a basis.
TheoremIf a vector space has a basis consisting of m vectors, then any
set of more than m vectors is linearly dependent.
S Li
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Span, Linear
Independence,
Dimension
Math 240
Spanning sets
Linearindependence
Bases andDimension
Dimension
Corollary
Any two bases for a single vector space have the same number
of elements.
Definition
The number of elements in any basis is the dimension
of thevector space. We denote it dim V .
Examples
1. dim Rn = n
2. dim M m×n(R) = mn3. dim P n = n + 1
4. dim P = ∞
5. dim C k(I ) = ∞6. dim{0} = 0
A vector space is called finite dimensional if it has a basis witha finite number of elements, or infinite dimensional otherwise.
S Li Di i
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Span, Linear
Independence,
Dimension
Math 240
Spanning sets
Linearindependence
Bases andDimension
Dimension
TheoremIf dim V = n, then any set of n linearly independent vectors in
V is a basis.
TheoremIf dim V = n, then any set of n vectors that spans V is a basis.
Corollary
If S is a subspace of a vector space V then
dim S ≤ dim V
and S = V only if dim S = dim V .