1. SPANNING

2. 3 6
= 3 1 2 3 6
is a multiple of 1 2 11 8 = 1 2 4 1 2 4
is a linear combination of 11 8
and

3. and
Is a linear combination of

4. B is a linear combination of A and C
VECTORS
SCALARS
The vectors form
a COMMUTATIVE
GROUP with an
identity called C
The scalars form
a FIELD with
identity for + called 0
identity for × called 1
2 A C = B
B is a linear combination of A and C

5. -1 -1 -1
Reduced echelon form

8. Every solution to this homogenious
system is a linear combination
of the vectors:

9. SPAN V. The system has infinitely many solutions that form
a vector space, V.
It would not be humanly
possible to list all of these
solutions.
Fortunately, the entire vector
space is determined by just
two vectors.
We say that these two vectors
SPAN V.
Every solution to this homogenious
system is a linear combination
of the vectors:

10. definition: A set of vectors S is said to span a vector space V
if every vector in V is a linear combination of the vectors in S.

11. and
SPAN the plane ( ie: R2 )

12. and
SPAN the plane ( ie: R2 )
Every point is a linear
combination of these
two vectors.
For example: =

13. and
SPAN the plane ( ie: R2 )
Every point is a linear
combination of these
two vectors.
For example: =