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6.2 Subspaces

Use the subspace test carefully, prove the first structural consequences, and learn the standard matrix and equation-defined examples that recur throughout linear algebra.

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MATH1030: Linear algebra I

Rigorous linear algebra notes on systems, matrices, structure, and proof, with interaction used only where it clarifies the mathematics.

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Chapter 1

Systems of equations

Learn to read equations as full solution sets.

1.1

1.1 Equations and solution sets

Chapter 2

Matrices and elimination

Build matrix intuition and use row reduction with purpose.

2.1

2.1 Matrix basics

2.2

2.2 Augmented matrices and row operations

2.3

2.3 Gaussian elimination and RREF

2.4

2.4 Solution-set types

Chapter 3

Matrix algebra

Matrix multiplication, transpose, and structural matrix notation.

3.1

3.1 Matrix multiplication and identity matrices

3.2

3.2 Transpose and special matrices

Chapter 4

Solution structure

Homogeneous systems, null spaces, and the shape of full solution sets.

4.1

4.1 Homogeneous systems and null space

Chapter 5

Invertibility

Understand when a matrix can be undone and why that matters.

5.1

5.1 Invertible matrices

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Chapter 6

Vector spaces

Move from matrix procedures to the structure of spaces, span, independence, and basis.

6.1

6.1 Vector spaces

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6.2

6.2 Subspaces

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6.3

6.3 Linear combinations and span

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6.4

6.4 Linear dependence and independence

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6.5

6.5 Basis and dimension

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6.6

6.6 Column space, row space, and rank

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Prerequisites

6.1

6.1 Vector spaces

Key terms in this unit

More notes in this series

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6.1 Vector spaces

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6.3 Linear combinations and span