Linear Dependence and Independence
Linear Combination
Given the numbers a1, a2, ..., an and the vectors v1, v2, ..., vn, a linear combination is each of the vectors of the form:
Given the vectors 
, calculate the linear combination vector 
Can the vector 
be expressed as a linear combination of the vectors 
?
Linearly Dependent Vectors
Vectors are linearly dependent if there is a linear combination of them that equals the zero vector, without the coefficients of the linear combination being zero.
Properties
1.If several vectors are linearly dependent, then at least one of them can be expressed as a linear combination of the others.
If a vector is a linear combination of others, then all the vectors are linearly dependent.
2.Two vectors in the plane are linearly dependent if, and only if they are parallel.
3.Two vectors in the plane 
= (u1, u2) and 
= (v1, v2) are linearly dependent if their components are proportional.
Linearly Independent Vectors
Several vectors are linearly independent if none of them can be expressed as a linear combination of the others.
a1 = a2 = ··· = an = 0
Examples
Determine if the vectors are linearly dependent or independent:
= (3, 1) and = (2, 3)
Linearly independent
Determine if the vectors are linearly dependent or independent:
= (x − 1, 3) and = (x + 1, 5)
Are vectors are linearly dependent for x = 4.
Determine if the vectors are linearly dependent or independent:
= (5, 3 − x ) and = (x + 9, 3x + 1)
They are linearly dependent for x = 1 and x = −22
Check that the line segment joining the midpoints of sides AB and AC of the triangle: A (3, 5), B (−2, 0), C (0, −3) are parallel to the side BC and equal to its half.
