University Linear Algebra Linear Transformations — Unit Notes
Every definition and theorem this unit expects you to know, stated once each and written to be revised from: 6 definitions and 9 theorems, 8 worked examples, 1 diagram, in 14 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.
Linear Transformations — what the notes cover
9 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.
Verifying that a map is linear
- Definition — Linear transformation
- Theorem — Linear maps preserve linear combinations
Also in this section: 1 worked example, 1 note on method and common errors.
The standard matrix of a linear transformation
- Theorem — The standard matrix
- Theorem — A linear map is fixed by its values on a basis
Also in this section: 1 worked example, 1 note on method and common errors.
Rotations, reflections, projections and shears
- Definition — The four geometric operators of the plane
- Theorem — Recognising the operators from the matrix
Also in this section: 1 worked example, 1 note on method and common errors, 1 diagram.
The kernel and range of a linear transformation
- Definition — Kernel and range
- Theorem — Kernel and range are subspaces
Also in this section: 1 worked example, 1 note on method and common errors.
The dimension theorem
- Definition — Rank and nullity of a linear transformation
- Theorem — The dimension theorem
Also in this section: 1 worked example, 1 note on method and common errors.
One-to-one and onto transformations
- Definition — One-to-one and onto
- Theorem — Testing one-to-one and onto
Also in this section: 1 worked example, 1 note on method and common errors.
Composition of linear transformations
- Theorem — Composition of linear transformations
Also in this section: 1 worked example, 1 note on method and common errors.
Invertible transformations and isomorphism
- Definition — Invertible transformation and isomorphism
- Theorem — Isomorphic spaces have equal dimension
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 Linear Transformations 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.
Linear Transformations practice worksheet About the Linear Transformations unit test