University Linear Algebra Projections, Least Squares and Symmetric Matrices — 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, 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.
Projections, Least Squares and Symmetric Matrices — 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.
Orthogonal projection onto a subspace
- Definition — Orthogonal projection onto a subspace
- Theorem — Orthogonal decomposition
- Theorem — Best approximation
Also in this section: 1 worked example, 1 note on method and common errors, 1 diagram.
The Gram-Schmidt process
- Theorem — The Gram-Schmidt process
Also in this section: 1 worked example, 1 note on method and common errors.
Least-squares solutions and the normal equations
- Definition — Least-squares solution
- Theorem — The normal equations
- Theorem — When the least-squares solution is unique
Also in this section: 1 worked example, 1 note on method and common errors.
Fitting a line or a curve
- Definition — Design matrix of a model
- Theorem — The least-squares line
Also in this section: 1 worked example, 1 note on method and common errors, 1 diagram.
Orthogonal matrices
- Definition — Orthogonal matrix
- Theorem — What an orthogonal matrix preserves
Also in this section: 1 worked example, 1 note on method and common errors.
Orthogonal diagonalization of a symmetric matrix
- Definition — Orthogonally diagonalizable
- Theorem — Eigenvectors of a symmetric matrix
- Theorem — The spectral theorem for real symmetric matrices
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 Projections, Least Squares and Symmetric Matrices 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.
Projections, Least Squares and Symmetric Matrices practice worksheet About the Projections, Least Squares and Symmetric Matrices unit test