How To Show Vectors Are Linearly Independent

In linear algebra, the concept of linear independence is fundamental. It describes the relationship between vectors in a vector space, determining whether any vector in a set can be written as a linear combination of the others. This article provides a comprehensive guide on how to demonstrate that vectors are linearly independent, covering definitions, methods, and practical examples.

Understanding Linear Independence

Before diving into the methods, it's crucial to grasp the definition of linear independence. A set of vectors {v₁, v₂, ..., vₙ} is said to be linearly independent if the only solution to the equation:

c₁v₁ + c₂v₂ + ... + cₙvₙ = 0

is the trivial solution where c₁ = c₂ = ... = cₙ = 0. In other words, the only way to obtain the zero vector as a linear combination of these vectors is by setting all the scalar coefficients to zero.

Conversely, if there exists a non-trivial solution (at least one cᵢ ≠ 0) to the equation above, the vectors are linearly dependent. This means at least one vector can be expressed as a linear combination of the others.

Methods to Prove Linear Independence

Several methods can be used to determine whether a set of vectors is linearly independent. The choice of method often depends on the specific vectors and the context of the problem. Here are the most common approaches:

1. The Definition Method (Setting up a Homogeneous System)
2. Using the Determinant (for n Vectors in Rⁿ)
3. Row Reduction (Gaussian Elimination)
4. Inspection (for Simple Cases)
5. Using Linear Transformations

1. The Definition Method (Setting up a Homogeneous System)

This method directly applies the definition of linear independence.

Steps:

• Set up the equation: Write the equation c₁v₁ + c₂v₂ + ... + cₙvₙ = 0, where v₁, v₂, ..., vₙ are the vectors in question and c₁, c₂, ..., cₙ are scalar coefficients.
• Convert to a system of linear equations: Express the vector equation as a system of linear equations by equating corresponding components.
• Solve the system: Find all possible solutions for c₁, c₂, ..., cₙ.
• Check for trivial solution:
  • If the only solution is c₁ = c₂ = ... = cₙ = 0, then the vectors are linearly independent.
  • If there exists a non-trivial solution (at least one cᵢ ≠ 0), then the vectors are linearly dependent.

Example:

Determine whether the vectors v₁ = (1, 2), v₂ = (3, 4) in R² are linearly independent.

• Set up the equation: c₁(1, 2) + c₂(3, 4) = (0, 0)
• Convert to a system of linear equations:
  • c₁ + 3c₂ = 0
  • 2c₁ + 4c₂ = 0
• Solve the system: Multiply the first equation by -2: -2c₁ - 6c₂ = 0. Add this to the second equation: -2c₂ = 0, which implies c₂ = 0. Substitute c₂ = 0 into the first equation: c₁ + 3(0) = 0, which implies c₁ = 0.
• Check for trivial solution: The only solution is c₁ = 0 and c₂ = 0. Therefore, the vectors v₁ and v₂ are linearly independent.

Advantages:

• Directly applies the definition.
• Works for vectors in any vector space (as long as you can perform scalar multiplication and vector addition).

Disadvantages:

• Can be computationally intensive for larger sets of vectors or more complex systems of equations.

2. Using the Determinant (for n Vectors in Rⁿ)

This method applies specifically when you have n vectors in Rⁿ (e.g., 2 vectors in R², 3 vectors in R³).

Steps:

• Form a matrix: Create a square matrix A whose columns (or rows) are the given vectors.
• Calculate the determinant: Compute the determinant of the matrix A, denoted as det(A) or |A|.
• Check the determinant:
  • If det(A) ≠ 0, then the vectors are linearly independent.
  • If det(A) = 0, then the vectors are linearly dependent.

Example:

Determine whether the vectors v₁ = (1, 2), v₂ = (3, 4) in R² are linearly independent.

• Form a matrix: A = [[1, 3], [2, 4]]
• Calculate the determinant: det(A) = (1 * 4) - (3 * 2) = 4 - 6 = -2
• Check the determinant: det(A) = -2 ≠ 0. Therefore, the vectors v₁ and v₂ are linearly independent.

Advantages:

• Relatively quick and efficient for smaller sets of vectors in Rⁿ.

Disadvantages:

• Only applicable when you have n vectors in Rⁿ.
• Calculating determinants for large matrices can still be computationally expensive.

3. Row Reduction (Gaussian Elimination)

Row reduction is a powerful technique that can be used to determine linear independence and solve systems of linear equations.

Steps:

• Form a matrix: Create a matrix A whose columns are the given vectors.
• Row reduce the matrix: Use elementary row operations (swapping rows, multiplying a row by a non-zero scalar, adding a multiple of one row to another) to transform the matrix A into its row echelon form (REF) or reduced row echelon form (RREF).
• Analyze the REF/RREF:
  • If the REF/RREF of A has a pivot (leading 1) in every column, then the vectors are linearly independent.
  • If the REF/RREF of A has a column without a pivot, then the vectors are linearly dependent. This means there is a free variable, and thus infinitely many solutions to the homogeneous equation.

Example:

Determine whether the vectors v₁ = (1, 2, 1), v₂ = (2, 5, 0), v₃ = (3, 7, 1) in R³ are linearly independent.

• Form a matrix: A = [[1, 2, 3], [2, 5, 7], [1, 0, 1]]

• Row reduce the matrix:

  • Subtract 2 times row 1 from row 2: [[1, 2, 3], [0, 1, 1], [1, 0, 1]]
  • Subtract row 1 from row 3: [[1, 2, 3], [0, 1, 1], [0, -2, -2]]
  • Add 2 times row 2 to row 3: [[1, 2, 3], [0, 1, 1], [0, 0, 0]]

• Analyze the REF: The matrix is now in row echelon form. Notice that the third column does not have a pivot. Therefore, the vectors v₁, v₂, and v₃ are linearly dependent.

Advantages:

• Works for any number of vectors in Rⁿ.
• Provides information about the linear dependencies if the vectors are dependent.
• Can be used to solve systems of linear equations.

Disadvantages:

• Can be computationally intensive for large matrices.
• Requires a good understanding of row operations.

4. Inspection (for Simple Cases)

In some simple cases, linear independence can be determined by inspection, without performing any calculations.

Cases where inspection is sufficient:

• The set contains the zero vector: If the set of vectors contains the zero vector, the vectors are always linearly dependent. This is because you can write a linear combination of the vectors where the coefficient of the zero vector is non-zero, and all other coefficients are zero, resulting in the zero vector.
  • Example: { (1, 2), (0, 0), (3, 4) } is linearly dependent.
• The set contains only one vector: A set containing only one non-zero vector is always linearly independent. The only way to get the zero vector is to multiply the vector by zero.
  • Example: { (1, 2) } is linearly independent.
• Two vectors, one is a scalar multiple of the other: If one vector is a scalar multiple of the other, the vectors are linearly dependent.
  • Example: { (1, 2), (2, 4) } is linearly dependent because (2, 4) = 2 * (1, 2).

Advantages:

• Very quick and easy for simple cases.

Disadvantages:

• Only applicable to a limited set of cases.

5. Using Linear Transformations

Linear transformations can sometimes be used to determine linear independence.

Theorem:

If T: V -> W is a one-to-one (injective) linear transformation, then the set {v₁, v₂, ..., vₙ} in V is linearly independent if and only if the set {T(v₁), T(v₂), ..., T(vₙ)} in W is linearly independent.

Steps:

• Find a suitable linear transformation: Choose a linear transformation T that simplifies the problem or maps the vectors to a space where it is easier to determine linear independence.
• Apply the transformation: Compute T(v₁), T(v₂), ..., T(vₙ).
• Determine linear independence in the new space: Use one of the other methods (definition, determinant, row reduction, inspection) to determine whether the transformed vectors {T(v₁), T(v₂), ..., T(vₙ)} are linearly independent in W.
• Apply the theorem:
  • If T is one-to-one and {T(v₁), T(v₂), ..., T(vₙ)} is linearly independent, then {v₁, v₂, ..., vₙ} is linearly independent.
  • If T is one-to-one and {T(v₁), T(v₂), ..., T(vₙ)} is linearly dependent, then {v₁, v₂, ..., vₙ} is linearly dependent.

Example:

Let V be the space of polynomials of degree at most 2, and let W = R³. Consider the vectors v₁ = 1 + x, v₂ = x + x², v₃ = 1 + x² in V. Determine whether these vectors are linearly independent.

• Find a suitable linear transformation: Define T: V -> R³ as T(p(x)) = (p(0), p(1), p(2)). This is a linear transformation. Furthermore, if T(p(x)) = (0, 0, 0), then p(0) = p(1) = p(2) = 0. Since p(x) is a polynomial of degree at most 2 with three distinct roots, it must be the zero polynomial. Thus, T is one-to-one.
• Apply the transformation:
  • T(v₁) = T(1 + x) = (1, 2, 3)
  • T(v₂) = T(x + x²) = (0, 2, 6)
  • T(v₃) = T(1 + x²) = (1, 2, 5)
• Determine linear independence in the new space: Form the matrix A = [[1, 0, 1], [2, 2, 2], [3, 6, 5]]. Row reduce this matrix:
  • Subtract 2 times row 1 from row 2: [[1, 0, 1], [0, 2, 0], [3, 6, 5]]
  • Subtract 3 times row 1 from row 3: [[1, 0, 1], [0, 2, 0], [0, 6, 2]]
  • Divide row 2 by 2: [[1, 0, 1], [0, 1, 0], [0, 6, 2]]
  • Subtract 6 times row 2 from row 3: [[1, 0, 1], [0, 1, 0], [0, 0, 2]] The RREF has a pivot in every column. Therefore, T(v₁), T(v₂), and T(v₃) are linearly independent.
• Apply the theorem: Since T is one-to-one and {T(v₁), T(v₂), T(v₃)} is linearly independent, then {v₁, v₂, v₃} is linearly independent.

Advantages:

• Can simplify the problem by mapping vectors to a more convenient space.

Disadvantages:

• Requires finding a suitable one-to-one linear transformation.
• May still require using other methods to determine linear independence in the new space.

Practical Considerations and Tips

• Choose the right method: Consider the specific vectors and the context of the problem when choosing a method. The determinant method is quick for n vectors in Rⁿ, while row reduction is more general.
• Be careful with calculations: Errors in calculations, especially when computing determinants or performing row reduction, can lead to incorrect conclusions. Double-check your work.
• Understand the geometric interpretation: Linear independence has a geometric interpretation. In R², two vectors are linearly independent if they do not lie on the same line through the origin. In R³, three vectors are linearly independent if they do not lie on the same plane through the origin.
• Practice: The more you practice, the more comfortable you will become with determining linear independence. Work through a variety of examples.

FAQ

• What does it mean if vectors are linearly dependent?

  If vectors are linearly dependent, it means that at least one vector can be written as a linear combination of the others. Geometrically, this means that the vectors are in some sense "redundant." For example, in R², two linearly dependent vectors lie on the same line through the origin.

• Can the zero vector be part of a linearly independent set?

  No, the zero vector cannot be part of a linearly independent set. Any set containing the zero vector is linearly dependent.

• Is the empty set linearly independent?

  Yes, the empty set is considered linearly independent by convention. There are no vectors in the set, so the condition for linear dependence (existence of a non-trivial linear combination that equals the zero vector) cannot be satisfied.

• How does linear independence relate to the rank of a matrix?

  The rank of a matrix is the number of linearly independent columns (or rows) of the matrix. If the columns of a matrix are linearly independent, then the rank of the matrix is equal to the number of columns.

Conclusion

Determining whether vectors are linearly independent is a crucial skill in linear algebra. By understanding the definition of linear independence and mastering the various methods for proving it, you can confidently tackle a wide range of problems. Remember to choose the method that is most appropriate for the specific vectors and context, and always double-check your calculations. From setting up homogeneous systems to utilizing determinants and row reduction, each technique offers a unique approach to unveiling the relationships between vectors. Furthermore, recognizing simple cases through inspection and leveraging linear transformations can provide efficient shortcuts. With practice and a solid understanding of these principles, you'll be well-equipped to analyze and solve linear independence problems effectively.