Lesson 6: Linear Dependence, Span & Basis

1. Linear Combination

A linear combination of vectors is any vector you can get by adding scaled versions of them.

Example:

Any vector in 2D can be written as a linear combination of \(\mathbf{i}\) and \(\mathbf{j}\).

2. Span of Vectors

The span of a set of vectors is the set of all possible linear combinations you can make from them.

Examples:

• Span of one non-zero vector → a line through the origin
• Span of two non-parallel vectors in 2D → the entire plane
• Span of three vectors in 3D → the entire space (if they are not coplanar)

3. Linear Dependence & Independence

A set of vectors is linearly dependent if one vector can be written as a linear combination of the others (redundant).

Linearly independent if none can be expressed as combination of others → each adds new direction.

Examples:

• \( \left\{\begin{pmatrix} 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \end{pmatrix}\right\} \) → independent
• \( \left\{\begin{pmatrix} 1 \\ 2 \end{pmatrix}, \begin{pmatrix} 2 \\ 4 \end{pmatrix}\right\} \) → dependent (second = 2 × first)

4. Basis & Dimension

A basis is a set of linearly independent vectors that span the whole space.

Dimension = number of vectors in a basis.

Examples:

• 2D plane → dimension 2, basis: standard i and j
• 3D space → dimension 3
• A line through origin → dimension 1

Every vector space has a basis (but we won’t prove it here).

Summary – Lesson 6

• Span = all possible linear combinations
• Linearly dependent = one vector is combination of others
• Basis = minimal spanning set that is independent
• Dimension = size of basis

Next steps: matrices, systems of equations, and transformations.