Basis

Basis#

Let’s dive into the concept of linear algebra basis using SageMath to illustrate.

Review of key concepts

  • Vector Space: A collection of vectors that can be added together and multiplied by scalars (usually real or complex numbers), while still obeying specific rules.

  • Linear Combination: Adding multiples of vectors together (e.g., 3vector1 + 2vector2).

  • Span: The set of all possible linear combinations you can create from a given set of vectors.

  • Linear Independence: A set of vectors is linearly independent if none of them can be expressed as a linear combination of the others.

Definition: Basis:

A linearly independent set of vectors that spans the entire vector space. It’s essentially a minimal set of vectors that can “build” any other vector in the space through linear combinations.

A basis is a special type of spanning set that is also linearly independent, making it the minimal and most efficient set of vectors needed to span the space.

SageMath Examples

Let’s use SageMath, a powerful open-source mathematics software system, to make these concepts more tangible.

1. Creating Vectors and Vector Spaces

v1 = vector([1, 2, 3])
v2 = vector([4, 5, 6])
v3 = vector([7, 8, 9])
V = VectorSpace(QQ, 3)  # Create a 3-dimensional vector space over rational numbers
print(V)

Vector space of dimension 3 over Rational Field

2. Checking Linear Independence

V.linear_dependence([v1, v2, v3])
# A non-empty list indicates linear dependence

[(1, -2, 1)]

3. Finding a Basis

A = matrix(QQ, [[1, 2, 3], [4, 5, 6], [7, 8, 9]])
row_space_basis = A.row_space()
basis_vectors = row_space_basis.basis()
print(basis_vectors)

[
(1, 0, -1),
(0, 1, 2)
]

SageMath finds a basis for the row space of matrix A.

This means these two vectors can be combined to create any row of A.

4. Expressing rows as a combination of the basis vectors

Each row of matrix \( A \) can be expressed as a combination of the basis vectors found by SageMath. Let’s start with your matrix \( A \):

\( A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \)

The basis vectors for the row space of \( A \) are given by:

\( \mathbf{v}_1 = (1, 0, -1) \) \( \mathbf{v}_2 = (0, 1, 2) \)

These basis vectors mean that any row of \( A \) can be written as a linear combination of \(\mathbf{v}_1\) and \(\mathbf{v}_2\).

Let’s express each row of \( A \) as such a combination.

Row 1: \( (1, 2, 3) \)

We want to find scalars \( c_1 \) and \( c_2 \) such that:

\( (1, 2, 3) = c_1 (1, 0, -1) + c_2 (0, 1, 2) \)

This leads to the system of equations:

$ \begin{cases} c_1 = 1 \ c_2 = 2 \

  • c_1 + 2c_2 = 3 \end{cases} $

Let’s solve it step-by-step:

  1. From the first equation: \( c_1 = 1 \).

  2. From the second equation: \( c_2 = 2 \).

  3. Check the third equation: \( -1 + 2 \cdot 2 = -1 + 4 = 3 \).

So, the combination for the first row is:

\( (1, 2, 3) = 1 \cdot (1, 0, -1) + 2 \cdot (0, 1, 2) \)

Let’s verify this:

vector([1, 2, 3]) == 1 * vector([1,0,-1]) + 2* vector([0,1,2]) 

True

The combinations for the second and third row have been calculated for you and are shown below:

  • \( (4, 5, 6) = 4 \cdot (1, 0, -1) + 5 \cdot (0, 1, 2) \)

  • \( (7, 8, 9) = 7 \cdot (1, 0, -1) + 8 \cdot (0, 1, 2) \)

In summary, the basis vectors \((1, 0, -1)\) and \((0, 1, 2)\) can indeed be combined using appropriate scalars to reconstruct each row of matrix \( A \).

Why Basis Matters#

  • Compact Representation: A basis gives you the most concise way to describe a vector space.

  • Coordinates: Every vector in the space gets a unique “address” (coordinates) with respect to a chosen basis.

  • Transformations: Basis vectors help understand how linear transformations (like rotations or reflections) work.

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