Summary - AP statistics Matrix algebra tutorial-Vector Independence

Independent vs. Dependent Vectors
One vector is dependent on other vectors, if it is a linear combination of the other vectors.


Linear Combination of Vectors
If one vector is equal to the sum of scalar multiples of other vectors, it is said to be a linear combination of the other vectors.

For example, suppose a = 2b + 3c, as shown below.

11 1 3 2*1 + 3*3
= 2 + 3 =
16 2 4 2*2 + 3*4

a b c 2b + 3c

Note that 2b is a scalar multiple and 3c is a scalar multiple. Thus, a is a linear combination of b and c.


Linear Dependence of Vectors
A set of vectors is linearly independent if no vector in the set is (a) a scalar multiple of another vector in the set or (b) a linear
combination of other vectors in the set; conversely, a set of vectors is linearly dependent if any vector in the set is (a) a scalar multiple
of another vector in the set or (b) a linear combination of other vectors in the set.

Consider the row vectors below.

a= 1 2 3 d= 2 4 6

b= 4 5 6 e= 0 1 0

c= 5 7 9 f= 0 0 1


Note the following:


Vectors a and b are linearly independent, because neither vector is a scalar multiple of the other.

Vectors a and d are linearly dependent, because d is a scalar multiple of a; i.e., d = 2a.

Vector c is a linear combination of vectors a and b, because c = a + b. Therefore, the set of vectors a, b, and c is linearly
dependent.

Vectors d, e, and f are linearly independent, since no vector in the set can be derived as a scalar multiple or a linear combination
of any other vectors in the set.


Test Your Understanding
Problem 1

Consider the row vectors shown below.

0 1 2 3 2 1
a b


3 3 3 3 4 5
c d