Linear Independence

Linear independence is a fundamental concept of linear algebra. It 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 a vector space, a set of vectors is said to be linearly independent if no vector in the set can be expressed as a linear combination of the other vectors in the set.

For example, in a two-dimensional vector space, the vectors (1, 0) and (0, 1) are linearly independent because no scalar multiple of one can produce the other.

However, the vectors (1, 2) and (2, 4) are linearly dependent because the second vector is simply twice the first.

Note: Each vector in the set provides specific or different directions of the space.

Mathematical Definition of Linear Independence

A set of vectors {v₁, v₂, . . . , v_n} is linearly independent if the equation:

c₁v₁ + c₂v₂ + . . . + c_nv_n = 0; has only the trivial solution c₁ = c₂ = . . . = c_n = 0.

In contrast, if there exist non-zero scalars c₁, c₂ . . . c_n such that the equation above holds, then the set of vectors is linearly dependent.

Criteria for Linear Independence

For a set of vectors v₁, v₂, ..., v_n in n-dimensional space, create a matrix M with these vectors as its columns. Then, calculate the determinant of M. If the determinant is

• Non-zero, the vectors are linearly independent.
• Zero, they are linearly dependent.

Note: If the set of vectors forms an orthogonal set (i.e., each pair of vectors in the set is orthogonal to each other), then they are linearly independent.

Steps to Determine Linear Independence

To 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 the 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.

A=  \begin{bmatrix} 1 & 2 & 3\\ 2 & -1 & 0 \\ 3 & 0 & 1 \\\end{bmatrix}

⇒ det(A) = 1[(-1)(1) - (0)(0)] - 2[(2)(1) - (3)(0)] + 3[(2)(0) - (-1)(3)]
⇒ det(A) = 1(-1) - 2(2) + 3(3)
⇒ det(A) = -1 - 4 + 9 = 4

Since det(A) ≠ 0, the vectors are linearly independent.

Linear Independence in Vector Spaces

Vectors are considered linearly independent if no vector in a set can be represented as a linear combination of the others. In other words, a set of vectors {v₁, v₂, . . . , v_n} is linearly independent if the only solution to the equation:

c₁v₁ + c₂v₂ + . . . + c_nv_n = 0

(where c₁, c₂ . . . c_n are scalars, not all zero) is the trivial solution where all scalars are zero.

Examples of Linear Independence in Vectors

Consider a set of vectors in ℝ³: {(1, 0, 0), (0, 1, 0), (0, 0, 1)}.

These vectors are linearly independent because no vector can be expressed as a linear combination of the others.

Now consider, {(1, 0, 0), (2, 0, 0), (3, 0, 0)}. It would be linearly dependent, as the third vector is a scalar multiple of the first.

Application of Linear Independence

Linear independence finds applications in various fields:

• Machine Learning: In machine learning, linearly independent features are crucial for avoiding multicollinearity issues in regression models, ensuring that each feature provides unique predictive information.

• Computer Graphics: Linear independence is essential in computer graphics for defining transformations and generating realistic visual effects.

• Physics and Engineering: In physics and engineering, linear independence forms the formulation and analysis of complex systems, from modeling physical phenomena to designing efficient structures and mechanisms.

How to Prove Linear Independence?

To prove linear independence, set up the equation c₁v₁ + c₂v₂ + . . . + c_nv_n = 0. Organize the vectors into a matrix, solve the system AC = 0, and determine if the only solution is c₁ = c₂ = . . . = c_n = 0. If so, the vectors are linearly independent; otherwise, they are dependent.

Problem: Determine if the vectors are linearly independent.

• v₁ : <1, 2>
• v₂: <3, 4>
• v₃: < 2, 5>

Solution:

We can construct a matrix with these vectors as columns and perform row reduction:

\begin{bmatrix} 1 & 3 & 2\\ 2 & 4 & 5 \\ \end{bmatrix}

Performing row reduction(R₂ → R₂ − 2R₁), we find:

\begin{bmatrix} 1 & 3 & 2\\ 0 & -2 & 1 \\\end{bmatrix}

The matrix has 2 pivot columns (columns 1 and 2), but there are 3 columns total. This means column 3 is a free variable, so the system has a non-trivial solution.

Therefore, the vectors are linearly dependent.

Also Check

• Linear Algebra
• List of Linear Algebra Symbols
• Matrices

Solved Questions

Question 1: Determine whether the vectors \vec{v}_1 = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, \quad\vec{v}_2 = \begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix}, \quad\vec{v}_3 = \begin{bmatrix} 7 \\ 8 \\ 9 \end{bmatrix} are linearly independent or linearly dependent.

Solution:

To check linear independence, we examine the equation:

c₁​v_1​ + c₂​v_2​ + c₃​v₃ ​= 0

Substitute the vectors:

c_1 \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} +c_2 \begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix} +c_3 \begin{bmatrix} 7 \\ 8 \\ 9 \end{bmatrix} =\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}\

This gives the system:

\begin{cases}c_1 + 4c_2 + 7c_3 = 0 \\2c_1 + 5c_2 + 8c_3 = 0 \\3c_1 + 6c_2 + 9c_3 = 0\end{cases}

Write the augmented matrix:

\left[\begin{array}{ccc|c}1 & 4 & 7 & 0 \\2 & 5 & 8 & 0 \\3 & 6 & 9 & 0\end{array}\right]

Apply Gaussian elimination:

R_2​ ← R_2​ − 2R_1​,
R₃ ← R₃ − 3R₁

\left[\begin{array}{ccc|c}1 & 4 & 7 & 0 \\0 & -3 & -6 & 0 \\0 & -6 & -12 & 0\end{array}\right]

R₃ ​← R₃ ​− 2R₂​

\left[\begin{array}{ccc|c}1 & 4 & 7 & 0 \\0 & -3 & -6 & 0 \\0 & 0 & 0 & 0\end{array}\right]

The last row is all zeros, indicating a free variable.

Since a non-trivial solution exists (not all constants are zero), the vectors are linearly dependent.

Question 2: Let x be a real number.

{v}_1 = \begin{bmatrix} 1 +x \\ x\end{bmatrix}, {v}_2 = \begin{bmatrix} 2 +x \\ x\end{bmatrix}\quad

Define the following 2×1 Are A₁ and A₂​ linearly independent?

Solution:

Case 1: x ≠ 0

From the second equation:

x = λx
⇒λ = 1

Plug into the first:

2+x = 1⋅(1+x)
⇒ 2+x = 1+x
⇒ False

Contradiction → So not linearly dependent in this case.

Case 2: x = 0

Then the vectors become:

A_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \quad A_2 = \begin{bmatrix} 2 \\ 0 \end{bmatrix}

So they are linearly dependent when x = 0.

Unsolved Question

Question 1: Determine whether the following vectors are linearly independent:

\vec{v}_1 = \begin{bmatrix} 1 \\ 2 \end{bmatrix}, \quad\vec{v}_2 = \begin{bmatrix} 3 \\ 6 \end{bmatrix}

Question 2: Let \vec{a} = \begin{bmatrix} 1 \\ x \end{bmatrix}, \quad\vec{b} = \begin{bmatrix} 2 \\ 2x \end{bmatrix}

For what values of x are \vec{a} and \vec{b} linearly independent?

Question 3: Are the following vectors linearly independent?

\vec{u}_1 = \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}, \quad\vec{u}_2 = \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix}, \quad\vec{u}_3 = \begin{bmatrix} 1 \\ 1 \\ 3 \end{bmatrix}

Question 4: Let

\vec{w}_1 = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, \quad\vec{w}_2 = \begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix}, \quad\vec{w}_3 = \begin{bmatrix} a \\ b \\ c \end{bmatrix}

For which values of a,b,c are vector \vec{w}_1 , \vec{w}_2,\vec{w}_3.