Linearly dependent and linearly independent vectors

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Linear combination of vectors - definition
Linearly independent vectors - definition
Linearly dependent vectors - definition
Linearly dependent vectors properties
Linearly dependent and linearly independent vectors examples

Definition. A linear combination of vectors a₁, ..., a_n with coefficients x₁, ..., x_n is a vector

x₁a₁ + ... + x_na_n.

Definition. A linear combination x₁a₁ + ... + x_na_n are called trivial if all the coefficients x₁, ..., x_n are zero.

Definition. A linear combination x₁a₁ + ... + x_na_n are called nontrivial if at least one of the coefficients x₁, ..., x_n is not zero.

Definition. The vectors a₁, ..., a_n are called linearly independent if there are no non-trivial combination of these vectors equal to the zero vector.

That is, the vector a₁, ..., a_n are linearly independent if x₁a₁ + ... + x_na_n = 0 if and only if x₁ = 0, ..., x_n = 0.

Definition. The vectors a₁, ..., a_n are called linearly dependent if there exists a non-trivial combination of these vectors is equal to the zero vector.

Linearly dependent vectors properties:

For 2-D and 3-D vectors.
Two linearly dependent vectors are collinear. (Collinear vectors are linearly dependent.)
For 3-D vectors.
Three linear dependence vectors are coplanar. (Three coplanar vectors are linearly dependent.)
For an n-dimensional vectors.
n + 1 vectors always linearly dependent.

Linearly dependent and linearly independent vectors examples:

Example 1. Check whether the vectors a = {3; 4; 5}, b = {-3; 0; 5}, c = {4; 4; 4}, d = {3; 4; 0} are linearly independent.

Solution:

The vectors are linearly dependent, since the dimension of the vectors smaller than the number of vectors.

Example 2. Check whether the vectors a = {1; 1; 1}, b = {1; 2; 0}, c = {0; -1; 1} are linearly independent.

Solution: Calculate the coefficients in which a linear combination of these vectors is equal to the zero vector.

x₁a + x₂b + x₃c₁ = 0

This vector equation can be written as a system of linear equations

	x₁ + x₂ = 0
	x₁ + 2x₂ - x₃ = 0
	x₁ + x₃ = 0

Solve this system using the Gauss method

1	1	0	~
1	2	-1
1	0	1

from 2 row we subtract the 1-th row;from 3 row we subtract the 1-th row:

~	1	1	0	0	~	1	1	0	~
	1 - 1	2 - 1	-1 - 0	0 - 0		0	1	-1
	1 - 1	0 - 1	1 - 0	0 - 0		0	-1	1

from 1 row we subtract the 2 row; for 3 row add 2 row:

~	1 - 0	1 - 1	0 - (-1)	0 - 0	~	1	0	1
	0	1	-1	0		0	1	-1
	0 + 0	-1 + 1	1 + (-1)	0 + 0		0	0	0

This solution shows that the system has many solutions, ie exist nonzero combination of numbers x₁, x₂, x₃ such that the linear combination of a, b, c is equal to the zero vector, for example:

-a + b + c = 0

means vectors a, b, c are linearly dependent.

Answer: vectors a, b, c are linearly dependent.

Example 3. Check whether the vectors a = {1; 1; 1}, b = {1; 2; 0}, c = {0; -1; 2} are linearly independent.

Solution: Calculate the coefficients in which a linear combination of these vectors is equal to the zero vector.

x₁a + x₂b + x₃c₁ = 0

This vector equation can be written as a system of linear equations

	x₁ + x₂ = 0
	x₁ + 2x₂ - x₃ = 0
	x₁ + 2x₃ = 0

Solve this system using the Gauss method

1	1	0	~
1	2	-1
1	0	2

from 2 row we subtract the 1-th row;from 3 row we subtract the 1-th row:

~	1	1	0	0	~	1	1	0	~
	1 - 1	2 - 1	-1 - 0	0 - 0		0	1	-1
	1 - 1	0 - 1	2 - 0	0 - 0		0	-1	2

from 1 row we subtract the 2 row; for 3 row add 2 row:

~	1 - 0	1 - 1	0 - (-1)	0 - 0	~	1	0	1	~
	0	1	-1	0		0	1	-1
	0 + 0	-1 + 1	2 + (-1)	0 + 0		0	0	1

from 1 row we subtract the 3 row; for 2 row add 3 row:

~	1 - 0	0 - 0	1 - 1	0 - 0	~	1	0	0
	0 + 0	1 + 0	-1 + 1	0 + 0		0	1	0
	0	0	1	0		0	0	1

This means that the system has a unique solution x₁ = 0, x₂ = 0, x₃ = 0, and the vectors a, b, c are linearly independent.

Answer: vectors a, b, c are linearly independent.

Vectors Vectors Definition. Main information Component form of a vector with initial point and terminal point Length of a vector Direction cosines of a vector Equal vectors Orthogonal vectors Collinear vectors Coplanar vectors Angle between two vectors Vector projection Addition and subtraction of vectors Scalar-vector multiplication Dot product of two vectors Cross product of two vectors (vector product) Scalar triple product (mixed product) Linearly dependent and linearly independent vectors Decomposition of the vector in the basis

Online calculators with vectors

Tasks and exercises with vector 2D

Tasks and exercises with vector 3D

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