136 lecture slides 2 post

Lecture 2

Content: Spanning

Math 136 Dan Wolczuk Faculty of Mathematics University of Waterloo

In linear algebra, sets of all possible linear combinations of aset of vectors are very important.

Definition

The set S of all possible linear combinations of a set of vectorsB = {v 1, . . . ,v k} in Rn is called the span of the set B andwe write

S = SpanB = Span{v 1, . . . ,v k}= {t1

v 1 + + tkv k | t1, . . . , tk R}

We also say that S is spanned by B and that B is a spanningset for S.


We will sometimes want to write a set differently. For a set{t1v 1 + + tkv k +b | t1, . . . , tk R

}we will just write

x = t1v 1 + + tkv k +b , t1, . . . , tk R

and call this a vector equation of the set.

Notice that the vector equation gives us the general form ofany vector in the set.


Example

Which of the following sets contains the vector x =12

1

.(a) S = Span

10

0

,01

0

,00

1

Solution:


Example


1

.(b) T = Span

11

1

,212

Solution:


Example


Example


1

.(c) U = Span

20

1

,10

1

Solution:


Example

Describe geometrically the set S = Span

{[11

]}in R2.

Solution:


Example

Describe geometrically the set S = Span

10

1

,01

0

in R3.Solution:


Example

Find a simplified vector equation of the following sets anddescribe the set geometrically.

(a) Span

12

1

,24

2

Solution:


Example


Example


(b) Span

11

2

,03

1

,12

3

Solution:


Example


What are some other possible simplified vector equations for

Span

11

2

,03

1

,12

3

?


Example


(c) Span

11

1

,00

0

Solution:


Example


(d) Span

10

1

,01

0

,00

1

Solution:


136 lecture slides 2 post

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