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:

Linear Combination

Linear Combination


Given the vectors Vectors, calculate the linear combination vector Linear Combination Vector


Can the vector Linear Combination Vector be expressed as a linear combination of the vectors Vectors?

Linear Combination Operations

Linear Combination Operations

Linear Combination Operations

Linear Combination Operations

Linear Combination Solution

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 u = (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:

Vector = (3, 1) and = (2, 3)

Linear Dependence Operations

Linearly independent


Determine if the vectors are linearly dependent or independent:

Vector = (x − 1, 3) and = (x + 1, 5)

formulas

Are vectors are linearly dependent for x = 4.


Determine if the vectors are linearly dependent or independent:

Vector = (5, 3 − x ) and = (x + 9, 3x + 1)

Linear Dependence Operations

Linear Dependence Solution

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.

Linearly Independent Vectors

Parallel Vectors

Vector Calculations

Vector Calculations

Vector Calculations

Proportional Components

Vector Calculations

Vector Calculations

Vector Calculations

Vector Calculations

Vector Solution