How To Find Basis For Subspace

How to Find a Basis for a Subspace

Finding a basis for 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. This article will guide you through various methods for determining a basis, catering to different levels of understanding and tackling diverse scenarios. We'll explore the process with detailed explanations, examples, and practical tips to solidify your understanding.

Understanding the Core Concepts

Before delving into the methods, let's review some key definitions:

What is a Vector Space?

A vector space is a collection of vectors that satisfy certain axioms under addition and scalar multiplication. These axioms ensure that vectors can be added together and multiplied by scalars (numbers) in a consistent and predictable way. Examples include Rⁿ (the set of all n-dimensional vectors with real entries) and the space of all polynomials of degree less than or equal to n.

What is a Subspace?

A subspace is a subset of a vector space that is itself a vector space. Crucially, it must be closed under addition and scalar multiplication. This means that if you add any two vectors within the subspace, the result remains within the subspace, and similarly, multiplying any vector in the subspace by a scalar keeps the result within the subspace. The zero vector must also be included in the subspace.

What is a Basis?

A basis for a subspace is a set of linearly independent vectors that span the subspace. Let's break down these terms:

• Linear Independence: A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the other vectors. 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.

• Spanning the Subspace: A set of vectors spans a subspace if every vector in the subspace can be expressed as a linear combination of the vectors in the set. This means that you can reach any point in the subspace by appropriately scaling and adding the vectors in the basis.

A basis is a minimal spanning set, meaning it contains the fewest number of vectors necessary to span the subspace. Any other spanning set will contain redundant vectors.

Methods for Finding a Basis

Now, let's explore the practical methods for determining a basis for a subspace:

Method 1: Using the Span of a Set of Vectors

This method is useful when the subspace is defined as the span of a set of vectors. The process involves determining if the vectors are linearly independent. If they are, they form a basis. If they are linearly dependent, you need to identify and remove the redundant vectors.

Steps:

1. Write the vectors as columns of a matrix: Arrange the given vectors as columns of a matrix A.

2. Perform Gaussian Elimination (Row Reduction): Apply row operations to transform the matrix into row echelon form or reduced row echelon form.

3. Identify Pivot Columns: The columns of the original matrix A that correspond to the pivot columns (columns with leading 1s in the row echelon form) form a basis for the subspace.

Example:

Let's find a basis for the subspace spanned by the vectors v₁ = (1, 2, 3), v₂ = (2, 4, 6), and v₃ = (0, 1, 1).

1. Matrix:

   A = | 1  2  0 |
       | 2  4  1 |
       | 3  6  1 |
   

2. Row Reduction: Applying Gaussian elimination, we might get (the specific steps depend on the chosen row operations):

   R = | 1  2  0 |
       | 0  0  1 |
       | 0  0  0 |
   

3. Pivot Columns: The first and third columns are pivot columns. Therefore, the vectors v₁ = (1, 2, 3) and v₃ = (0, 1, 1) form a basis for the subspace. v₂ is a multiple of v₁ (v₂ = 2v₁), hence it's redundant.

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

The null space of a matrix A, denoted as Null(A), is the set of all vectors x such that Ax = 0. To find a basis for the null space, we use Gaussian elimination and solve the homogeneous system of linear equations.

Steps:

1. Row Reduce the Matrix: Transform the matrix A into reduced row echelon form.

2. Express Free Variables: Identify the pivot variables (corresponding to pivot columns) and the free variables (corresponding to columns without pivots).

3. Solve for Pivot Variables: Express the pivot variables in terms of the free variables.

4. Construct Basis Vectors: For each free variable, assign it a value of 1 and set the other free variables to 0. Solve for the pivot variables based on your expressions from step 3. The resulting vectors form a basis for the null space.

Example:

Find a basis for the null space of the matrix:

A = | 1  2 -1 |
    | 2  4 -2 |
    | 3  6 -3 |

1. Row Reduction: Reducing A to reduced row echelon form, we get:

   R = | 1  2  -1 |
       | 0  0   0 |
       | 0  0   0 |
   

2. Free Variable: x₂ is the free variable (no pivot in the second column).

3. Solve for Pivot Variable: From the first row, x₁ = -2x₂ + x₃

4. Basis Vectors:

   • Set x₂ = 1, x₃ = 0: x₁ = -2, x₂ = 1, x₃ = 0. This gives the vector (-2, 1, 0).
   • Set x₂ = 0, x₃ = 1: x₁ = 1, x₂ = 0, x₃ = 1. This gives the vector (1, 0, 1).

Thus, {(-2, 1, 0), (1, 0, 1)} forms a basis for the null space of A.

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. A basis for the column space can be found using Gaussian elimination.

Steps:

1. Row Reduce the Matrix: Reduce the matrix A to row echelon form or reduced row echelon form.

2. Identify Pivot Columns: The columns of the original matrix A that correspond to the pivot columns in the row echelon form form a basis for the column space.

Example:

Find a basis for the column space of the matrix from the previous example:

A = | 1  2 -1 |
    | 2  4 -2 |
    | 3  6 -3 |

1. Row Reduction: We already know the reduced row echelon form has a pivot only in the first column.

2. Pivot Column: Only the first column is a pivot column. Therefore, {(1, 2, 3)} forms a basis for the column space of A.

Method 4: Basis for a Subspace Defined by a System of Linear Equations

If the subspace is defined by a system of homogeneous linear equations, you can find a basis by finding the null space of the coefficient matrix. This is essentially Method 2.

Steps:

1. Form the Coefficient Matrix: Create a matrix A where each row corresponds to a linear equation in the system.

2. Find the Null Space: Use Method 2 to find a basis for the null space of A. This basis will be a basis for the subspace defined by the system of equations.

Method 5: Subspace defined by conditions (e.g., symmetric matrices)

Sometimes subspaces are defined by specific conditions imposed on the vectors, like the set of all symmetric matrices of a certain size. In such cases, you need to cleverly construct a basis based on those conditions. This often involves understanding the structure of the elements of the subspace.

Example: Symmetric 2x2 matrices

A symmetric 2x2 matrix has the form:

| a  b |
| b  c |

We can express this as a linear combination of basis matrices:

a| 1  0 | + b| 0  1 | + c| 0  0 |
  | 0  0 |   | 1  0 |   | 0  1 |

Therefore, a basis for the subspace of symmetric 2x2 matrices is:

{ | 1  0 | , | 0  1 | , | 0  0 | }
  | 0  0 |   | 1  0 |   | 0  1 |

Important Considerations and Advanced Techniques

• Uniqueness of Basis: While the dimension of a subspace is unique (the number of vectors in any basis), the specific basis vectors are not unique. Different methods might yield different bases for the same subspace.

• Dimension of a Subspace: The number of vectors in a basis for a subspace is equal to the dimension of the subspace. This is a crucial invariant property.

• Orthogonal Bases: For some applications, it's advantageous to find an orthogonal basis (where vectors are mutually perpendicular) or an orthonormal basis (where vectors are orthogonal and have unit length). The Gram-Schmidt process is a standard method for orthogonalizing a given basis.

• Change of Basis: It's possible to change from one basis to another using a change-of-basis matrix. This is valuable in various linear algebra applications.

• Eigenspaces: In eigenvalue problems, eigenspaces (subspaces spanned by eigenvectors corresponding to a particular eigenvalue) are particularly important. Finding bases for these eigenspaces is crucial for understanding the properties of linear transformations.

Finding a basis for a subspace is a cornerstone skill in linear algebra. By understanding the underlying concepts and mastering the methods outlined above, you'll be well-equipped to tackle various problems in linear algebra and related fields. Remember to practice regularly and work through diverse examples to solidify your understanding and develop your problem-solving abilities. The more you practice, the more intuitive these procedures will become.