MAST10007 Chap.12 Inner Products, Projections and Gram–Schmidt
Inner Products, Projections and Gram–Schmidt
Chapter 12 develops inner products, projections and gram–schmidt for University of Melbourne MAST10007. An inner product generalises the dot product and creates length, angle, and orthogonality. Projection splits a vector into a component inside a subspace and an orthogonal residual. Gram–Schmidt converts an independent list into an orthogonal or orthonormal basis without changing its span.
At each stage subtract projections onto every earlier orthogonal direction, then normalise only after a nonzero residual remains. It pairs exact definitions with a new completed example, two concept diagrams, diagnostic contrasts and answered retrieval prompts.
Projection answers a decomposition question: which component lies in the chosen subspace, and which residual is perpendicular to it?
The correct formula depends on whether the available basis is merely orthogonal or already orthonormal. Gram–Schmidt constructs such a basis sequentially, preserving the span at every stage while removing components along earlier directions. Each new residual should be checked for orthogonality before normalisation.
These checks make the geometry visible and prevent a memorised formula from hiding a missing denominator or an omitted projection.
The orthogonal-complement language unifies these checks. The residual belongs to the complement of the target subspace, while the projection belongs to the subspace itself. Their sum reconstructs the original vector and their inner product vanishes.
This gives a compact audit with distinct roles: membership for the projected component, orthogonality for the residual, and reconstruction for the decomposition. When all three hold, the geometry and algebra support the same conclusion.
What this chapter covers
- 01
Inner products
- 02
Norm and distance
- 03
Orthogonality
- 04
Cauchy–Schwarz
- 05
Projection onto a vector
- 06
Projection onto a subspace
- 07
Orthogonal complements
- 08
Gram–Schmidt
- 09
Orthonormal coordinates
- 10
Stability and checking
- 11
Residual orthogonality as a projection check
- 12
Orthogonal versus orthonormal projection formulas
- 13
Span preservation through Gram–Schmidt
Inner Products, Projections and Gram–Schmidt worked example
- stepSet u1=v1=(1,1,0), with u1·u1=2.
- stepCompute proj_{u1}(v2)=((v2·u1)/(u1·u1))u1=(1/2)(1,1,0).
- stepSubtract to get u2=v2-proj=(1/2,-1/2,1). The dot product u1·u2 is zero.
- stepNormalise if requested: e1=u1/√2 and e2=u2/√(3/2). The unnormalised pair already forms an orthogonal basis for the original span.
Key terms
- Inner products
- An inner product takes two vectors to a scalar and satisfies conjugate symmetry, linearity in the chosen argument convention, and positive definiteness. In real coordinate spaces the standard example is the dot product.
- Norm and distance
- The norm induced by an inner product is ||v||=sqrt(<v,v>), and distance is ||u−v||. Positive definiteness guarantees a nonzero vector has positive length. Normalisation divides by this norm.
- Orthogonality
- Orthogonality means <u,v>=0. An orthogonal list of nonzero vectors is automatically independent because taking the inner product with one list vector isolates its coefficient.
- Cauchy–Schwarz
- Cauchy–Schwarz bounds |<u,v>| by ||u||||v||. It guarantees the cosine formula stays within its valid range and leads to the triangle inequality. Equality means the vectors are linearly dependent.
- Projection onto a vector
- Projection onto nonzero u uses <v,u>/<u,u> times u under the stated convention. The coefficient is chosen so v−proj_u(v) is orthogonal to u. This orthogonal residual characterises the closest point on the line.
- Projection onto a subspace
- For an orthonormal basis q1,…,qk of W, proj_W(v)=Σ<v,qi>qi. With an orthogonal but unnormalised basis, each term needs division by <qi,qi>. Mixing the two formulas loses scale.
- Gram–Schmidt process
- Gram–Schmidt takes an independent ordered list and repeatedly removes projections onto earlier orthogonal directions. The resulting nonzero vectors are mutually orthogonal and span the same successive subspaces. Normalising them produces an orthonormal basis.
- Orthogonal residual
- For projection onto a subspace, the residual is the original vector minus its projected component. It must be orthogonal to every direction in the target subspace. Dotting the residual with a basis is therefore a direct verification.
Inner Products, Projections and Gram–Schmidt FAQ
How do I check inner products?
An inner product takes two vectors to a scalar and satisfies conjugate symmetry, linearity in the chosen argument convention, and positive definiteness. In real coordinate spaces the standard example is the dot product. ((4)/2)u=(2,2).
What is the main trap in inner products, projections and gram–schmidt?
Projection onto u requires u≠0 and divides by u·u, not ||u||. Gram–Schmidt subtracts every previous projection, not only the most recent one.
Can AI help with this MAST10007 topic?
Yes. Ask Sia to explain one step, create a fresh practice problem, or check your reasoning. Use it to learn rather than complete graded assessment, and follow University of Melbourne academic-integrity rules.
Why must Gram–Schmidt subtract projections onto every earlier direction?
Each new vector can contain components along all previously constructed orthogonal directions. Removing only the most recent component can leave the residual nonorthogonal to earlier vectors. The ordered sum of projections removes every component already represented.
How do I check a projection without recomputing the whole formula?
Confirm that the projected vector lies in the target subspace and that the residual is orthogonal to each basis direction of that subspace. Together these properties characterise the orthogonal decomposition and expose a missing scale factor.
Exam move
Use the varied 11-page chapter as a dependency map: define each object, reproduce the fresh worked example, explain both diagrams, answer the two retrieval prompts, and perform an independent check. For inner products, projections and gram–schmidt, revisit the first line where your representation, dimensions, or theorem conditions diverge.
Label the target subspace, its basis, the projected component, and the residual before calculating. Decide whether the basis is orthogonal or orthonormal and select the matching formula. During Gram–Schmidt, test each new residual against every earlier direction and track that the span has not changed. Finish by reconstructing the original vector as projection plus residual.
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