Span, Linear Independence, and Dimension
Span, Linear
Independence,
Dimension
Math 240
Spanning sets
Linear
independence
Bases and
Dimension
Span, Linear Independence, and Dimension
Math 240 ¡ª Calculus III
Summer 2013, Session II
Thursday, July 18, 2013
Span, Linear
Independence,
Dimension
Agenda
Math 240
Spanning sets
Linear
independence
Bases and
Dimension
1. Spanning sets
2. Linear independence
3. Bases and Dimension
Span, Linear
Independence,
Dimension
Recap of span
Math 240
Spanning sets
Linear
independence
Bases and
Dimension
Yesterday, we saw how to construct a subspace of a vector
space as the span of a collection of vectors.
Question
What¡¯s the span of v1 = (1, 1) and v2 = (2, ?1) in R2 ?
Answer: R2 .
Today we ask, when is this subspace equal to the whole vector
space?
Span, Linear
Independence,
Dimension
Definition
Math 240
Spanning sets
Linear
independence
Bases and
Dimension
Definition
Let V be a vector space and v1 , . . . , vn ¡Ê 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 .
Theorem
Let v1 , . . . , vn be vectors in Rn . 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 ¡Ê Rn .
Span, Linear
Independence,
Dimension
Example
Math 240
Spanning sets
Linear
independence
Bases and
Dimension
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
?
?
? ?
1 ?2
4
c1
1 ?3? and x = ?c2 ? ,
A = ??1
4
3
5
c3
for an arbitrary v ¡Ê 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, the
span of these three vectors is a plane; they do not span R3 .
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