University Linear Algebra Rank and the Fundamental Subspaces — Unit Notes
Every definition and theorem this unit expects you to know, stated once each and written to be revised from: 5 definitions and 8 theorems, 6 worked examples, 2 diagrams, in 13 printable pages. This page lists what the notes cover. The notes themselves come with the University Linear Algebra bundle.
Unit notes — in the bundle
Not sold separately. The practice worksheet for this unit is free, with no email and no account.
Rank and the Fundamental Subspaces — what the notes cover
7 sections, in the order the unit is taught. Each entry below is the title of a definition or a theorem the notes state in full.
What you need
Also in this section: 1 note on method and common errors.
Solution sets in parametric vector form
- Definition — Parametric vector form
- Theorem — The solution set of a consistent system
Also in this section: 1 worked example, 1 note on method and common errors, 1 diagram.
The null space of a matrix
- Definition — Null space
- Theorem — A basis for the null space
Also in this section: 1 worked example, 1 note on method and common errors.
The column space and the row space
- Definition — Column space and row space
- Theorem — Membership in the column space
- Theorem — Row operations and the three spaces
Also in this section: 1 worked example, 1 note on method and common errors.
Bases for the null, column and row spaces
- Theorem — Bases for the three spaces
Also in this section: 1 worked example, 1 note on method and common errors.
Rank, nullity and the rank theorem
- Definition — Rank and nullity
- Theorem — The rank theorem
- Definition — The four fundamental subspaces
Also in this section: 1 worked example, 1 note on method and common errors, 1 diagram.
Rank and the consistency of a linear system
- Theorem — Consistency test by rank
- Theorem — Consistency for every right-hand side
Also in this section: 1 worked example, 1 note on method and common errors.
Where the notes are
The notes are one of the documents in the University Linear Algebra Solutions Bundle, beside the unit test, the challenge problems and every worked answer key. The Rank and the Fundamental Subspaces practice worksheet is free and stays free.
Every University Linear Algebra topic — the complete Solutions Bundle
One download, one payment, the whole program. For all 9 University Linear Algebra units: the worksheet, the reference notes, the challenge set, the unit test and every answer key — these notes included.
- Worked solutions, not answer lists — every step written out
- Covers the whole year's program at this level
- Less than the price of one hour of tutoring — for the entire year's solutions
Practise it, then test yourself
Notes on their own do not make anything stick. This unit has a free, printable practice worksheet, and an end-of-unit test that comes with the bundle.
Rank and the Fundamental Subspaces practice worksheet About the Rank and the Fundamental Subspaces unit test