MAT 242 Chap.11 Orthogonal Vectors and the Gram-Schmidt Algorithm
Orthogonal Vectors and the Gram-Schmidt Algorithm
Perpendicular in any dimension
The dot product measures angles, and the course needs only right angles: two vectors are orthogonal when their dot product is zero. Written with the transpose, u·v = uᵀv, which makes dot products part of matrix algebra.
Length is the square root of a vector's dot product with itself.
Orthogonal complements
The vectors perpendicular to every vector of a subspace W form another subspace, the orthogonal complement. Being perpendicular to a basis of W is enough, so the complement is the null space of the transpose of the basis matrix.
Reducing the basis matrix without transposing is the standard error here, and the paid chapter walks through a complete example in R⁴.
Orthogonal bases and Gram-Schmidt
If a basis is orthogonal, each coordinate is a single ratio of dot products, with no inverse required; on an ordinary basis that shortcut gives wrong numbers.
Gram-Schmidt converts any basis into an orthogonal one: keep the first vector, and from each later vector subtract its components along the new vectors already built. The course writes this with OLD and NEW labels.
Dividing each new vector by its length then gives an orthonormal basis, and the paid chapter carries three vectors in R⁴ through the whole process with a figure of the subtraction step.
The chain the course draws is worth remembering: spanning set, basis by pivot columns, orthogonal basis by Gram-Schmidt, orthonormal basis by normalising.
What the Final asks
The Final Exam skills list includes coordinates with respect to an orthogonal basis, a basis for an orthogonal complement, Gram-Schmidt in the OLD and NEW notation, and an orthonormal basis.
Sample solutions give only partial credit for running Gram-Schmidt when the question asks merely to normalise an already orthogonal basis, and for dividing vectors by their lengths before they are orthogonal. Reading the command word decides which of the two procedures is required.
Why orthogonal bases are worth the effort
With an ordinary basis, each coordinate requires solving a system or inverting a matrix.
With an orthogonal basis, each coordinate is one ratio of dot products, and with an orthonormal basis it is a single dot product. The same simplification carries over to projections, which is why the course builds orthogonal bases before treating projection with an arbitrary basis in the next chapter. On the Final, the same saving shows up as fewer lines of arithmetic and fewer places to slip.
What this chapter covers
- 01
Dot product, length and orthogonality
- 02
The orthogonal complement as a null space
- 03
Coordinates in an orthogonal basis
- 04
The Gram-Schmidt algorithm
- 05
Orthonormal bases
Worked example · free
Gram-Schmidt on two vectors
- 1NEW₁ = (1, 2, 2), with NEW₁·NEW₁ = 9.
- 3OLD₂·NEW₁ = 11, so NEW₂ = (3, 0, 4) − (11/9)(1, 2, 2) = (16/9, −22/9, 14/9).
- 1Scale NEW₂ by 9/2 to (8, −11, 7) and check: 8 − 22 + 14 = 0.
Key terms
- Orthogonal Vectors
- Orthogonal vectors are vectors whose dot product is zero.
- Orthogonal Complement
- The orthogonal complement of W is the subspace of all vectors perpendicular to every vector in W.
- Orthonormal Basis
- An orthonormal basis is an orthogonal basis whose vectors all have length 1.
- Gram-Schmidt Algorithm
- The Gram-Schmidt algorithm turns a basis into an orthogonal basis for the same subspace by removing components along earlier vectors.
Orthogonal Vectors and the Gram-Schmidt Algorithm FAQ
When can coordinates be found with dot products instead of an inverse?
Only when the basis is orthogonal. Each coordinate is then the dot product of the basis vector with u divided by that basis vector's dot product with itself; on a non-orthogonal basis the formula gives the wrong answer.
Why transpose the basis matrix when finding an orthogonal complement?
Perpendicularity to each basis vector is one equation whose coefficients are that vector's entries, so the basis vectors must be the rows of the system. Transposing puts them there.
Can I normalise vectors before running Gram-Schmidt?
It is allowed but makes the arithmetic much messier. The course recommends finding the orthogonal basis first and dividing each vector by its length only at the end.
Exam move
Run Gram-Schmidt on three vectors twice: once by hand with fractions, once with the calculator for dot products. Compare, and practise writing the final orthonormal basis in factored form. Write the OLD and NEW formula from memory before each practice session; most errors come from mixing an OLD vector into a denominator.
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