University Linear Algebra Basis, Dimension and Coordinates — Unit Notes
Every definition and theorem this unit expects you to know, stated once each and written to be revised from: 5 definitions and 9 theorems, 7 worked examples, 1 diagram, in 11 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.
Basis, Dimension and Coordinates — what the notes cover
6 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.
Bases and the standard bases
- Definition — Basis of a vector space
- Definition — The standard bases
- Theorem — Unique representation
- Theorem — A basis test for n vectors in n-space
Also in this section: 1 worked example, 1 note on method and common errors.
The dimension of a vector space
- Theorem — Every basis has the same size
- Definition — Dimension
- Theorem — Counting facts in an n-dimensional space
Also in this section: 1 worked example, 1 note on method and common errors.
Reducing a spanning set and extending an independent set
- Theorem — Removing a redundant vector
- Theorem — Extending an independent set
Also in this section: 2 worked examples, 1 note on method and common errors.
Coordinate vectors relative to a basis
- Definition — Coordinate vector relative to a basis
- Theorem — Coordinates respect linear combinations
- Theorem — Coordinates in n-space by solving a system
Also in this section: 1 worked example, 1 note on method and common errors, 1 diagram.
Coordinates in polynomial and matrix spaces
- Definition — Standard coordinates in polynomial and matrix spaces
- Theorem — Deciding questions in n-space
Also in this section: 2 worked examples, 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 Basis, Dimension and Coordinates 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.
Basis, Dimension and Coordinates practice worksheet About the Basis, Dimension and Coordinates unit test