Linear Independence Calculator

This online Linear Independence Calculator helps you to determine the linear independency and dependency between vectors. It is a very important idea in linear algebra that involves understanding the concept of the independence of vectors. In this article, we break down what dependent and independent variables are and explain how to determine if vectors are linearly independent?

What are Linear Dependence and Independence?

In vector spaces, if there is a nontrivial linear combination of vectors that equals zero, then the set of vectors is said to be linearly dependent. A vector is said to be linear independent when a linear combination does not exist. If the equation is $ a_1 * v_1 + a_2 * v_2 + a_3 * v_3 + a_4 * v_4 + \dots + a_{n-1} * v_{n-1} + a_n * v_n = 0 $, then the $v_1, v_2, v_3, v_4 \dots, v_{n-1}$ are linearly independent vectors.

Here zero (0) is the vector with in all coordinates holds if and only if $a_1 + a_2 + a_3 + a_4 + \dots + a_{n-1} + a_n = 0$.

Otherwise, we can say that vectors are linearly dependent. The only linear vector combination that provides the zero vector is known as trivial.

For example, v = (2, -1), then also take e₁ = (1, 0), e₂ = (0, 1)

Then 1 * e₂ + (-2) * e₁ + 1 * v = 1 * (0, 1) + (-2) * (1, 0) + 1 * (2, -1) = (0, 1) + (-2, 0) + (2, -1) = (0, 0) , so, we found a non-trivial combination of the vectors that provides zero. Hence, they are linearly dependent. Also, we can see that the e₁ and e₂

without problematic vector v are linearly independent vectors.

However, an online Wronskian Calculator will help you to determine the Wronskian of the given set of functions.

How to Check for Linear Dependence?

To check for linear dependence, we change the values from vector to matrices. For example, three vectors in two-dimensional space: $v(a_1, a_2), w(b_1, b_2), u(c_1, c_2)$, then write their coordinates as one matric with each row corresponding to the one of vectors.

\[ M = \begin{bmatrix} a_1 & a_2 \\ b_1 & b_2 \\ c_1 & c_2 \end{bmatrix} \]

Then matrix rank is equal to the maximal number of independent vectors among w, v, and u.

How to Tell if Vectors are Linearly Independent?

In order to check if vectors are linearly independent, the online linear independence calculator can tell about any set of vectors, if they are linearly independent. If you want to check it manually, then the following examples can help you for a better understanding.

Example 1

Find the values of h for which the vectors are linearly dependent, if vectors:

h₁ = 1, 1, 0
h₂ = 2, 5, -3
h₃ = 1, 2, 7

in 3 dimensions, then find they are linear independent or not?

Solution:

The vectors A, B, C are linearly dependent, if their determinant is zero. i.e. |D|=0

$$ A = (1, 1, 0), B = (2, 5, -3), C = (1, 2, 7) $$

$$ |D| = \begin{vmatrix} 1 & 1 & 0 \\ 2 & 5 & -3 \\ 1 & 2 & 7 \end{vmatrix} $$

$$ |D| = 1 \times \begin{vmatrix} 5 & -3 \\ 2 & 7 \end{vmatrix} - (1) \times \begin{vmatrix} 2 & -3 \\ 1 & 7 \end{vmatrix} + (0) \times \begin{vmatrix} 2 & 5 \\ 1 & 2 \end{vmatrix} $$

$$ |D| = 1 \times ((5) \times (7) - (-3) \times (2)) - (1) \times ((2) \times (7) - (-3) \times (1)) + (0) \times ((2) \times (2) - (5) \times (1)) $$

$$ |D| = 1 \times ((35) - (-6)) - (1) \times ((14) - (-3)) + (0) \times ((4) - (5)) $$

$$ |D| = 1 \times (41) - (1) \times (17) + (0) \times (-1) $$

$$ |D| = (41) - (17) + (0) $$

$$ |D| = 24 $$

$$ |D| = 24 ≠ 0 $$

Example 2

Determine if the columns of the matrix form a linearly independent set, when three-dimensions vectors are:

v₁ = (1, 1, 1)
v₂ = (1, 1, 1)
v₃ = (1, 1, 1)

then determine if the vectors are linearly independent.

Solution:

If their determinant is zero. i.e. |D|=0, then check for linear independence vectors A, B, C.

$$ A = (1, 1, 1), B = (1, 1, 1), C = (1, 1, 1) $$

$$ |D| = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{vmatrix} $$

$$ |D| = 1 \times |1111| - (1) \times |1111| + (1) \times |1111| $$

$$ |D| = 1 \times \begin{vmatrix} 1 & 1 \\ 1 & 1 \end{vmatrix} - (1) \times \begin{vmatrix} 1 & 1 \\ 1 & 1 \end{vmatrix} + (1) \times \begin{vmatrix} 1 & 1 \\ 1 & 1 \end{vmatrix} $$

$$ |D| = 1 \times ((1) - (1)) - (1) \times ((1) - (1)) + (1) \times ((1) - (1)) $$

$$ |D| = 1 \times (0) - (1) \times (0) + (1) \times (0) $$

$$ |D| = (0) - (0) + (0) $$

$$ |D| = 0 $$

Since |D|= 0, So vectors A, B, C are linearly dependent.

However, an online Jacobian Calculator allows you to find the determinant of the set of functions and the Jacobian matrix.

How does Linear Independence Calculator Works?

An online linear dependence calculator checks whether the given vectors are dependent or independent by following these steps:

Input:

Choose the number of vectors and coordinates from the drop-down list.
Substitute the given values or you can add random values in all fields by hitting the “Generate Values” button.
Click on the "CALCULATE" button.

Output:

The linearly independent calculator first tells the vectors are independent or dependent.
Then, the linearly independent matrix calculator finds the determinant of vectors and provide a comprehensive solution.

FAQs

How to check if vectors are linearly independent?

If the determinant of vectors A, B, C is zero, then the vectors are linear dependent. Apart from this, if the determinant of vectors is not equal to zero, then vectors are linear dependent.

How to know if a matrix is linearly independent?

Initially, we need to get the matrix into the reduced echelon form. If we get an identity matrix, then the given matrix is linearly independent.

The Bottom Line:

Use this online linear independence calculator to determine the determinant of given vectors and check all the vectors are independent or not. If there are more vectors available than dimensions, then all vectors are linearly dependent. Undoubtedly, finding the vector nature is a complex task, but this recommendable calculator will help the students and tutors to find the vectors dependency and independency.

References

Wikipedia: Linear Independence
Libre Texts: Linear Independence and the Wronskian
Cornell University: Linear Independence of Functions
Lumen Learning: Linear Independence in Functions