Introduction to Linear Independence Calculator With Steps:
Linear independence calculator is an online tool that is designed to determine whether the set of vectors is linearly independent or not. It checks the linear independence condition if a linear combination of vectors is equal to zero only when all coefficients of the combination are zero.
Linear independence checker is a valuable tool for students, professionals, and anyone who is working on vector spaces in mathematics and related fields.
What is Linear Independence?
Linear algebra is a process in linear algebra, where a set of vectors {v1, v2,…, vn} in a vector space is said to be linearly independent if a number of vectors in the set can be defined as a linear combination of other sets.
Linear independence of a vector can be written as:
$$ c_1 v_1 + c_2 v_2 + … + c_n v_n \;=\; 0 $$
Where c1, c2,…, cn are scalar of the given vector, and it has only the trivial solution c1 = c2 = … = cn = 0.
How to Calculate Linear Independence?
To determine linear independence, utilize our linear independence calculator to automate the process. Alternatively, you can manually assess by starting with a set of vectors and applying various methods to ascertain if they are linearly independent. Follow these steps to understand how to verify the independence of a given set of vectors.
Step 1: Consider a set of vectors {v1, v2,…, vn}. These vectors are linearly independent of the equation: c1v1 + c2v2 + … + cnvn = 0. This scalar has the trivial solution c1 = c2 = … = cn = 0
Step 2: Make a matrix A in the form of a column matrix. For example, if v1,v2,…, and vnv are vectors in Rm, then A will be an m × n matrix.
$$ A \;=\; \left( \begin{matrix} | & | & … & | \\ v_1 & v_2 & … & v_n \\ | & | & … & | \\ \end{matrix} \right) $$
Step 3: To check for linear independence, solve the equation Ac=0, where c=(c1c2….,cn) is the vector of coefficients and 0 is the zero vector.
$$ c \;=\; \left( \begin{matrix} c_1 \\ c_2 \\ \vdots \\ c_n \\ \end{matrix} \right) $$
You can use the Gaussian elimination row reduction or determinant method to solve the given system.
Step 4: If the solution of the matrix is equal to zero as Ac = 0, then the vectors {v1, v2,…, vn} are linearly independent. If the solution is non-trivial (i.e., c ≠ 0), then the vectors are linearly dependent.
Practical Example of a Linear Independent:
Let's see an example of the linear independence of a vector with a solution to know about this concept effortlessly.
Example: Find the linear independence of a given set of vector:
$$ \left[ \left( \begin{matrix} 1 \\ 1 \\ -2 \\ \end{matrix} \right),\; \left( \begin{matrix} 1 \\ -1 \\ 2 \\ \end{matrix} \right),\; \left( \begin{matrix} 3 \\ 1 \\ 4 \\ \end{matrix} \right) \right] $$
Solution:
Suppose x , y and z are the given vector to form equation as x+y+z=0.
$$ x \left( \begin{matrix} 1 \\ 1 \\ -2 \\ \end{matrix} \right) + y \left( \begin{matrix} 1 \\ -1 \\ 2 \\ \end{matrix} \right) + z \left( \begin{matrix} 3 \\ 1 \\ 4 \\ \end{matrix} \right) \;=\; \left( \begin{matrix} 0 \\ 0 \\ 0 \\ \end{matrix} \right) $$
Make a matrix A using the element of vector x, y and z.
$$ \left( \begin{matrix} 1 & 1 & 3 \\ 1 & -1 & 1 \\ -2 & 2 & 4 \\ \end{matrix} \right) $$
Use the gauss elimination method to convert the above matrix into reduced echelon form,
$$ \left[ \begin{matrix} 1 & 1 & 3 \\ 1 & -1 & 1 \\ -2 & 2 & 4 \\ \end{matrix} \right] $$
$$ R_2 \leftarrow R_2 - R_1 $$
$$ =\; \left[ \begin{matrix} 1 & 1 & 3 \\ 0 & -2 & -2 \\ -2 & 2 & 4 \\ \end{matrix} \right] $$
$$ R_3 \leftarrow R_3 + 2 \times R_1 $$
$$ =\; \left[ \begin{matrix} 1 & 1 & 3 \\ 0 & -2 & -2 \\ 0 & 4 & 10 \\ \end{matrix} \right] $$
$$ R_2 \leftarrow R_2 \div -2 $$
$$ =\; \left[ \begin{matrix} 1 & 1 & 3 \\ 0 & 1 & 1 \\ 0 & 4 & 10 \\ \end{matrix} \right] $$
$$ R_3 \leftarrow R_2 - 4 \times R_2 $$
$$ =\; \left[ \begin{matrix} 1 & 1 & 3 \\ 0 & 1 & 1 \\ 0 & 0 & 6 \\ \end{matrix} \right] $$
$$ R_3 \leftarrow R_3 - 4 \times R_2 $$
$$ R_3 \leftarrow R_3 \div 6 $$
$$ =\; \left[ \begin{matrix} 1 & 1 & 3 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\ \end{matrix} \right] $$
$$ R_2 - R_3\; and\; R_1 - R_2 $$
And lastly after R1-2R3
$$ \left( \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \right) $$
So x = y = z = 0 the only solution is the trivial solution then the given set of vectors is linearly independent.
How to Use the Linearly Independent Calculator?
Linear independent calculator have an easy-to-use interface, so you can easily use it to evaluate the linear independence of a given set solution. Before adding the input for the solutions of given vector space problems, you must follow some simple steps. These steps are:
- Add the number of vectors for the linearity of vectors.
- Add the size of the vector for the linearity of vectors.
- Enter the elements of the given vector in a set in the input box
- Review your input value for the vector before hitting the calculate button to start the calculation process in the linearly independent matrix calculator
- Click on the “Calculate” button to get the desired result of your given linear independent vector problem.
- If you want to try out our linear independence matrix calculator to check its accuracy in solution then use the load example.
- Click on the “Recalculate” button to get a new page for solving more linear combination questions.
Final Result of the Linear Independence Checker:
Linear independence of functions calculator gives you the solution to a given vector set problem when you add the input to it. It provides you with solutions . It may contain as:
- Result Option:
You can click on the result option and it provides you with a solution of vector set linearly independent questions.
- Possible Step:
When you click on the possible steps option it provides you with the solution of a linear independent problem where all calculation steps are included in detail.
Benefits of Using Linearly Independent Vectors Calculator:
Linear independence calculator with steps gives you multiple benefits whenever you use it to calculate linear independence problems and to get its solution immediately. These benefits are:
- Our linearly independent calculator saves the time and effort that you consume in solving complex vector space questions in a few seconds.
- It is a free-of-cost tool that provides you with a solution of a given vector to find its linear independence without paying a single penny.
- It is an adaptive tool that allows you to find the linear independence from the given vectors in two or maybe three dimensions in a linear independent calculator.
- You can use this linearly independent matrix calculator for practice so that you get a strong hold about this concept.
- Linear independence matrix calculator is a trustworthy tool that provides you with the accurate solutions as per your input to calculate the Linear Independent problem.
