11
Linear dependence and independence
Definition: A finite set S = cfw_x1 , x2 , . . . , xm of vectors in Rn is said to be linearly dependent if there exist scalars (real numbers) c1 , c2 , . . . , cm , not all of which are 0, such that
c1 x1 + c2 x2 + .

Chapter 29: Magnetic Fields due to Currents
Chapter 29 is a difficult chapter for many students. Why is this so? Remembering back
to electrostatics, in deriving E-fields (over chapters 21, 22, 23), we used two methods to
determine the fields: Coulombs met

12
12.1
Basis and dimension of subspaces
The concept of basis
Example: Consider the set
1
2
0
,
.
,
S=
2
1
1
Then span(S) = R2 . (Exercise). In fact, any two of the elements of S span R2 . (Exercise).
So we can throw out any one of them, for example,

10
Subspaces
(Now, we are ready to start the course . . . .)
Definitions:
A linear combination of the vectors v1 , v2 , . . . , vm is any vector of the form c1 v1 +
c2 v2 + . . . + cm vm , where c1 , . . . , cm R.
A subset V of Rn is a subspace if, when

13
13.1
The rank-nullity (dimension) theorem
Rank and nullity of a matrix
Definition: The nullity of the matrix A is the dimension of the null space of A, and is
denoted by N (A). (This is to be distinguished from Null(A), which is a subspace; the nullity