MAST10007 Chap.5 Vector Spaces and Subspaces
Vector Spaces and Subspaces
Chapter 5 develops vector spaces and subspaces for University of Melbourne MAST10007. A vector space is defined by operations and axioms, not by arrows. Polynomials, matrices, and functions can behave like vectors because addition and scalar multiplication satisfy the same rules. For a subset of a known vector space, use the subspace test: include the zero vector and prove closure under addition and scalar multiplication.
Nonhomogeneous constraints usually produce affine sets rather than subspaces. It pairs exact definitions with a new completed example, two concept diagrams, diagnostic contrasts and answered retrieval prompts.
Subspace reasoning should be written from the defining operations, not from a picture or a list of familiar examples. The zero-vector check is an efficient filter, but passing it is not a complete proof.
Closure statements must start with arbitrary members of the proposed set and show that their sum and scalar multiples satisfy the defining condition. Homogeneous linear constraints naturally support these steps, while a shifted constraint can fail immediately. This proof pattern prepares the language used later for spans, kernels, images, and eigenspaces.
What this chapter covers
- 01
Why abstract vector spaces
- 02
Vector-space axioms
- 03
Zero and additive inverses
- 04
Polynomial spaces
- 05
Matrix and function spaces
- 06
Subspace test
- 07
Solution sets of homogeneous systems
- 08
Counterexamples to closure
- 09
Affine sets versus subspaces
- 10
Proof-writing for subspaces
- 11
Using arbitrary elements in a closure proof
- 12
Homogeneous constraints and the zero vector
- 13
Separating affine sets from linear subspaces
Vector Spaces and Subspaces worked example
- stepThe zero vector satisfies 0-0+0=0, so W is nonempty and contains zero.
- stepIf u and v satisfy the equation, linearity gives the constraint on u+v as zero plus zero.
- stepFor any scalar c, the constraint on cu is c times zero, again zero.
- stepAll three conditions hold, so W is a subspace. Equivalently, W is the null space of the row matrix [1,-2,1].
Key terms
- Why abstract vector spaces
- Abstract vector spaces retain addition and scalar multiplication while changing what a vector looks like. Coordinates, polynomials, matrices, and functions can all qualify. The axioms guarantee that linear combinations behave consistently across these settings.
- Vector-space axioms
- The axioms include closure, associativity, commutativity of addition, additive identity and inverses, and distributive and associative scalar laws. In a familiar ambient space, these are inherited; a subset question is usually handled by the shorter subspace test.
- Zero and additive inverses
- The zero vector is the additive identity of the space and is unique. A subspace must contain it. Additive inverses follow from scalar closure because −u=(−1)u, but zero membership should still be checked explicitly in a clean proof.
- Polynomial spaces
- Polynomials of degree at most n form a vector space because sums and scalar multiples stay within the degree bound. Polynomials of degree exactly n do not: cancellation can lower degree, and the zero polynomial has no exact positive degree.
- Matrix and function spaces
- Matrices of a fixed size form a vector space under entrywise addition and scalar multiplication. Function spaces do too when the defining conditions are linear. A nonlinear condition such as f(0)=1 usually excludes the zero function.
- Subspace test
- For W inside a known vector space, test zero membership, addition closure, and scalar closure. A compressed alternative proves au+bv belongs to W for every u,v in W and scalars a,b, provided nonemptiness is established.
- Homogeneous constraint
- A homogeneous linear constraint has zero on its right-hand side. Its solution set contains the zero vector and is closed under linear combinations, so it forms a subspace of the relevant ambient vector space. This is the same mechanism behind null spaces.
- Affine set
- An affine set is a translated subspace. It can look like a line or plane while failing to contain the zero vector, so geometric shape alone does not prove subspace status. The defining equation and closure behaviour decide the classification.
Vector Spaces and Subspaces FAQ
How do I check why abstract vector spaces?
Abstract vector spaces retain addition and scalar multiplication while changing what a vector looks like. Coordinates, polynomials, matrices, and functions can all qualify. The axioms guarantee that linear combinations behave consistently across these settings. No; (0,0) does not satisfy the equation.
What is the main trap in vector spaces and subspaces?
Checking only that zero belongs is insufficient. A set through the origin may still fail closure. A translated line or plane that misses zero is affine, not a subspace.
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 is checking the zero vector not a complete subspace proof?
Zero membership only rules out sets that are shifted away from the origin. A proposed set can contain zero and still fail closure under addition or scalar multiplication. A complete proof must establish the operation rules for arbitrary members and scalars.
How should I write a closure argument without testing selected examples?
Begin with arbitrary vectors satisfying the defining condition. Apply the condition to their sum and to an arbitrary scalar multiple, simplify using linearity, and show the result still satisfies the condition. Examples may suggest the claim, but arbitrary elements prove it.
Exam move
Use the varied 9-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 vector spaces and subspaces, revisit the first line where your representation, dimensions, or theorem conditions diverge.
Practise the subspace test as a proof template with explicit quantifiers: state the ambient space, choose arbitrary members, test zero membership, addition, and scalar multiplication, then conclude. Contrast a homogeneous constraint with a shifted version and identify the first failed condition.
Repeat the argument for polynomial, matrix, and function examples so the reasoning depends on operations rather than coordinate appearance.
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