Orthogonal Projections
Closest point in a subspace.
Key idea
Closest point in a subspace.
In depth
Projection onto a subspace with basis in the columns of A is P = A(AᵀA)⁻¹Aᵀ. It satisfies P² = P and Pᵀ = P. It finds the nearest point in the subspace.
Worked example
Projection onto the line spanned by (1, 1): P = ½[1 1; 1 1].
Formula
Real-world uses
- Finding the closest achievable result, such as the nearest point in a model to a measured value
Common mistake
Forgetting that the result must be in the subspace, with error orthogonal to it.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Orthogonal Projections?
Closest point in a subspace. It is a topic at the expert level in Linear Algebra.
What is the formula for Orthogonal Projections?
The key formula is P = A(AᵀA)⁻¹Aᵀ.
Where is Orthogonal Projections used in real life?
Real-world examples: Finding the closest achievable result, such as the nearest point in a model to a measured value.
What should I learn before Orthogonal Projections?
Make sure you are comfortable with Linear Regression via Matrices, the previous topic in Linear Algebra.
What is a common mistake with Orthogonal Projections?
Forgetting that the result must be in the subspace, with error orthogonal to it.
How can I practise Orthogonal Projections?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Orthogonal Projections?
Move on to Orthogonal Matrices next.
Related expert topics in Linear Algebra
- Triple Product – Scalar triple product gives parallelepiped volume.
- Homogeneous Systems – Ax = 0 always has the trivial solution.
- Cofactor Expansion – Compute determinants recursively.
- Properties of Determinants – Multiplicative; row swaps change sign.
- Geometric Meaning of Determinants – Area or volume scaling factor.
- Cramer’s Rule for Systems – Solve systems using determinants.