Linear Algebra · Lesson 11
Orthogonality & Projections
Orthogonality generalizes perpendicularity to any number of dimensions. Projections let us find the closest point in a subspace to a given vector — the mathematical foundation of least squares regression, signal decomposition, and compression algorithms.
Quick Check
1. The projection of b=(6,2) onto a=(1,0) is:
(6,2)
(6,0)
(1,0)
(0,2)
2. The error vector e = b − proj_a b is always:
Parallel to a
Orthogonal to a
Equal to b
Zero
3. For an orthogonal matrix Q, Q⁻¹ equals:
Q
Qᵀ
−Q
Q²
4. An orthonormal set requires vectors that are:
Parallel to each other
Perpendicular and each of magnitude 1
All pointing in the same direction
Eigenvectors of the same matrix