How To Find Basis Of Subspace

How to Find the Basis of a Subspace

Finding the basis of a subspace is a fundamental concept in linear algebra. A basis provides a minimal set of vectors that can be used to generate every other vector within that subspace. Understanding how to find a basis is crucial for many applications, from solving systems of linear equations to understanding the structure of vector spaces. This comprehensive guide will walk you through various methods, providing clear explanations and illustrative examples.

Understanding Key Concepts

Before diving into the methods, let's solidify our understanding of crucial terms:

Vector Space:

A vector space is a collection of vectors that satisfy specific axioms under addition and scalar multiplication. These axioms ensure closure under these operations – meaning performing the operations on vectors within the space always results in another vector within that same space.

Subspace:

A subspace is a subset of a vector space that is itself a vector space. This means it must also satisfy the vector space axioms. Crucially, a subspace must contain the zero vector. Any subspace can be completely described by a basis.

Basis:

A basis of a vector space (or subspace) is a linearly independent set of vectors that spans the entire space. This means:

• Linear Independence: No vector in the basis can be written as a linear combination of the other vectors in the basis.
• Spanning Set: Every vector in the subspace can be expressed as a linear combination of the vectors in the basis.

Linear Combination:

A linear combination of vectors is a sum of scalar multiples of those vectors. For example, if we have vectors v₁, v₂, and v₃, a linear combination would be: c₁**v₁** + c₂**v₂** + c₃**v₃**, where c₁, c₂, and c₃ are scalars (numbers).

Linear Dependence/Independence:

A set of vectors is linearly dependent if at least one vector can be expressed as a linear combination of the others. Conversely, a set of vectors is linearly independent if none of the vectors can be written as a linear combination of the others. This can be determined using techniques like row reduction (Gaussian elimination).

Methods for Finding a Basis of a Subspace

Several methods exist to find a basis for a subspace. The most common approaches involve using matrices and row reduction. Let's explore these in detail:

Method 1: Using Spanning Sets and Row Reduction

This is the most prevalent method, particularly when the subspace is defined by a spanning set of vectors.

Steps:

1. Form a Matrix: Create a matrix where each column is one of the vectors from the spanning set.

2. Row Reduce: Perform Gaussian elimination (row reduction) to obtain the row echelon form (REF) or reduced row echelon form (RREF) of the matrix.

3. Identify Pivot Columns: The columns in the original matrix that correspond to the pivot columns (leading 1's) in the REF/RREF form a basis for the subspace.

Example:

Let's say our subspace is spanned by the vectors:

v₁ = [1, 2, 3], v₂ = [4, 5, 6], v₃ = [7, 8, 9]

1. Matrix Formation:

[ 1  4  7 ]
[ 2  5  8 ]
[ 3  6  9 ]

2. Row Reduction: After row reducing (using Gaussian elimination), you might obtain something similar to (the exact form depends on the row operations used):

[ 1  0  -1 ]
[ 0  1   2 ]
[ 0  0   0 ]

3. Pivot Columns: The first two columns are pivot columns. Therefore, the basis for the subspace is {v₁, v₂} = {[1, 2, 3], [4, 5, 6]}. v₃ is a linear combination of v₁ and v₂, as indicated by the zero row in the RREF.

Method 2: Finding a Basis for the Null Space (Kernel) of a Matrix

The null space (or kernel) of a matrix A is the set of all vectors x such that A**x = 0. Finding a basis for the null space is essential in many linear algebra problems.

Steps:

1. Augmented Matrix: Create an augmented matrix [A | 0], where 0 is a column vector of zeros.

2. Row Reduction: Perform row reduction to obtain the RREF.

3. Express Free Variables: Identify the free variables (variables corresponding to columns without leading 1's). Express the basic variables (variables corresponding to pivot columns) in terms of the free variables.

4. Basis Vectors: For each free variable, create a vector where the free variable is set to 1, and the other free variables are set to 0. The basic variables are then determined using the expressions derived in step 3. This set of vectors forms a basis for the null space.

Example:

Let's say we have the matrix:

A = [ 1  2  3 ]
    [ 4  5  6 ]

1. Augmented Matrix:

[ 1  2  3 | 0 ]
[ 4  5  6 | 0 ]

2. Row Reduction: (After row reduction)

[ 1  0  -1 | 0 ]
[ 0  1   2 | 0 ]

3. Free Variables: The third column doesn't have a pivot, so x₃ is a free variable.

4. Basis Vectors:

• Set x₃ = 1: From the RREF, we get x₁ = x₃ = 1 and x₂ = -2x₃ = -2. So, one basis vector is [1, -2, 1].

Therefore, the basis for the null space of A is {[1, -2, 1]}.

Method 3: Finding a Basis for the Column Space (Image) of a Matrix

The column space of a matrix A is the span of its column vectors. Finding a basis for the column space is similar to the first method.

Steps:

1. Form the Matrix: The matrix A itself represents the set of vectors you need to consider.

2. Row Reduction: Perform row reduction on the matrix A to the RREF.

3. Pivot Columns: The columns in the original matrix A corresponding to the pivot columns in the RREF form a basis for the column space.

Example:

Using the matrix A from the previous example:

A = [ 1  2  3 ]
    [ 4  5  6 ]

1. Row Reduction: (After row reduction) (The result will be similar to previous examples but using A without augmentation).

2. Pivot Columns: The first two columns are pivot columns.

3. Basis: The basis for the column space of A is {[1, 4], [2, 5]}. Note that this basis is constructed from the original matrix A's columns corresponding to the pivots.

Method 4: Subspace Defined by a Set of Linear Equations

If the subspace is defined by a set of homogeneous linear equations (equations equal to zero), you can use the following approach:

1. Form the Coefficient Matrix: Create a matrix whose rows are the coefficients of the equations.

2. Find the Null Space: Find the basis of the null space of the coefficient matrix using Method 2. This null space is the subspace defined by the equations.

Example:

Consider the subspace defined by:

x + 2y + 3z = 0 4x + 5y + 6z = 0

1. Coefficient Matrix:

[ 1  2  3 ]
[ 4  5  6 ]

2. Null Space: Follow Method 2 to find the basis of the null space of this matrix. This will yield the basis for the subspace defined by the given equations.

Important Considerations and Advanced Topics

• Dimension: The number of vectors in a basis is called the dimension of the subspace. This is an important invariant property of a subspace.

• Uniqueness: While the dimension of a subspace is unique, the basis itself is not unique. Many different sets of linearly independent vectors can span the same subspace.

• Basis for the Intersection of Subspaces: Finding a basis for the intersection of two or more subspaces requires more advanced techniques, often involving solving systems of linear equations simultaneously and carefully selecting linearly independent vectors.

• Orthogonal Basis: In some applications, it's beneficial to find an orthogonal basis (where all vectors are mutually orthogonal – their dot product is zero). Gram-Schmidt orthogonalization process is used for this purpose.

This comprehensive guide provides a solid foundation for finding the basis of a subspace. By mastering these methods and understanding the underlying concepts, you can tackle a wide range of problems in linear algebra and its various applications. Remember that practice is key to developing fluency and proficiency in these techniques. Work through numerous examples, varying the types of subspaces and the methods used, to deepen your understanding.