Steps to Determine Linear Independence

To o check for linear independence using matrices:

Step 1: Form a matrix where each column corresponds to one of the vectors

The first step in determining linear independence is to form a coefficient matrix using the given vectors. Let’s denote the given vectors as v₁, v₂, …, v_n, where each v_i is a column vector.

Step 2: Perform row operations to simplify the matrix then calculate determinant.

Once we have formed the coefficient matrix A, the next step is to calculate its determinant.

The determinant of a matrix is a scalar value that provides important information about the matrix. For an n × n matrix, the determinant can be computed using various methods such as cofactor expansion, Gaussian elimination, or using properties of determinants

Step 3: If the resulting matrix has a non-zero determinant, the vectors are linearly independent.

• If the determinant is non-zero, then the vectors are linearly independent.
• If the determinant is zero, then the vectors are linearly dependent.

If the determinant is non-zero, it implies that the system of equations represented by the coefficient matrix has only the trivial solution (where all coefficients are zero), indicating that the vectors are linearly independent.

On the other hand, if the determinant is zero, it suggests the existence of non-trivial solutions, indicating linear dependence among the vectors.

Example: Consider the following set of vectors in R³:

• v₁ : <1, 2, 3>
• v₂: 2, -1, 0>
• v₃: <3, 0, 1>

We want to determine whether these vectors are linearly independent.

Solution:

To test for linear independence, we’ll form a matrix where each column represents one of the vectors.

[Tex]A=  \begin{bmatrix} 1 & 2 & 3\\ 2 & -1 & 0 \\ 3 & 0 & 1 \\\end{bmatrix}[/Tex]

⇒ det (A) = 1[(−1)(-1) – 0] −2[(2)(1) − (3)(0)] + 3[(2)(0) − (−1)(3)]

⇒ det (A) = 1(1) − 2(2)+3(3)

⇒ det (A) = 1 − 4 + 9 = 6

Since the determinant of matrix A is non-zero (6 in this case), the vectors are linearly independent.

Linear independence is a fundamental concept in mathematics that has numerous applications in fields like physics, engineering, and computer science. It is necessary for determining the size of a vector space and finding solutions for optimization problems.

In this article, we will learn about linear independence, providing a simple explanation of its applications. We will understand the necessary steps for testing linear independence, their significance in the context of vector spaces and matrices as well.