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Linear Independence Calculator

Linear Independence Calculator

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An 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 \( a_1 * v_1 + a_2 * v_2 + a_3 * v_3 + a_4 * v_4 + … + a_{n – 1} * v_{n – 1} + a_n * v_n = 0 \), then the \( v_1, v_2, v_3, v_4, … , v_{n – 1}, v_n \) are linearly independent vectors.

Here zero (0) is the vector with in all coordinates holds if and only if \( a_1 + a_2 + a_3 + a_4 + … + a_{n-1} + a_n = 0 \). 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 \( e_1 = (1, 0), e_2 = (0, 1) \). Then \( 1 * e_2 + (-2) * e_1 + 1 * v = 1 * (0, 1) + (-2) * (1, 0) + 1 * (2, -1) = (0, 1) + (-2 ,0) + (2, -1) = (0, 0) \), so, we found a non-trivial combination of the vectors that provides zero. Hence, they are linearly dependent. Also, we can see that the \( e_1 and e_2 \) 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: \( v (a_1, a_2), w (b_1, b_2), v (c_1, c_2) \), then write their coordinates as one matric with each row corresponding to the one of vectors.

&& M = |D|= \left|\begin{array}{ccc}a_1 & a_1 & \\b_1 & b_2\\c_1 & c_2\end{array}\right| $$
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 \( h_1 = {1, 1, 0}, h_2 = {2, 5, -3}, h_3 = {1, 2, 7} \) 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

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

$$ |D|= \left|\begin{array}{ccc}1 & 1 & 0\\2 & 5 & -3\\1 & 2 & 7\end{array}\right| $$

$$|D|= 1 \times \left|\begin{array}{cc}5 & -3\\2 & 7\end{array}\right| – (1) \times \left|\begin{array}{cc}2 & -3\\1 & 7\end{array}\right| + (0) \times \left|\begin{array}{cc}2 & 5\\1 & 2\end{array}\right|$$

$$ |D|= 1 × ((5) × (7) − (−3) × (2)) − (1) × ((2) × (7) − ( − 3) × (1)) + (0) × ((2) × (2) − (5) × (1)) $$

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

$$ |D|=1 × (41) − (1) × (17) + (0) × (− 1) $$

$$ |D = (41) − (17) + (0) $$

$$ |D|= 24 $$

$$|D|= 24 ≠ 0 $$

Since |D|≠ 0, So vectors A, B, C are linearly independent.

Example 2:

Determine if the columns of the matrix form a linearly independent set, when three-dimensions vectors are \( v_1 = {1, 1, 1}, v_2 = {1, 1, 1}, v_3 = {1, 1, 1} \), 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.

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

$$ |D|= \left|\begin{array}{ccc}1 & 1 & 1\\1 & 1 & 1\\1 & 1 & 1\end{array}\right| $$

$$ |D|=1×|1111|−(1)×|1111|+(1)×|1111| $$

$$|D|= 1 \times \left|\begin{array}{cc}1 & 1\\1 & 1\end{array}\right| – (1) \times \left|\begin{array}{cc}1 & 1\\1 & 1\end{array}\right| + (1) \times \left|\begin{array}{cc}1 & 1\\1 & 1\end{array}\right|$$

$$ |D| = 1 × ((1) − (1)) − (1) × ((1) − (1)) + (1) × ((1) − (1)) $$

$$ |D|= 1 × (0) − (1) × (0) + (1) × (0) $$

$$ |D| = (0) − (0) + (0) $$

$$ |D|= 0 $$

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:

  • First, choose the number of vectors and coordinates from the drop-down list.
  • Now, 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.

Reference:

From the source of Wikipedia: Evaluating Linear independence, Infinite case, The zero vector, Linear dependence and independence of two vectors, Vectors in R2.

From the source of Libre Text: Linear Independence and the Wronskian, determinant of the corresponding matrix, linear differential equations, Affine independence.

From the source of Cornell University: Linear independence of values of G-functions, Alternative method using determinants, More vectors than dimensions, Natural basis vectors, Linear independence of functions, Space of linear dependencies.

From the source of Lumen Learning: Independent variable, Linear independence of functions, Space of linear dependencies, Affine independence.