--- title: "Vector Spaces and Subspaces" description: "Strip vectors down to the two operations and the axioms they satisfy, so that polynomials, matrices and functions count as vectors too. A subspace is a subset closed under both operations." canonical: https://lightmysky.com/learn/mathematics/vector-spaces-and-subspaces-mt_Zrddx-E6_n source: https://lightmysky.com/learn/mathematics/vector-spaces-and-subspaces-mt_Zrddx-E6_n.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Vector Spaces and Subspaces Strip vectors down to the two operations and the axioms they satisfy, so that polynomials, matrices and functions count as vectors too. A subspace is a subset closed under both operations. Subject: Mathematics · Area: Linear Algebra · Ages 18 to 19 Page: https://lightmysky.com/learn/mathematics/vector-spaces-and-subspaces-mt_Zrddx-E6_n ## Ready when they can - Check the closure conditions to decide whether a subset is a subspace - Give an example of a vector space whose elements are not arrows - Show that the solutions of a homogeneous system form a subspace ## Lesson: Spaces that behave like arrows A vector space is a set with an addition and a scaling that obey the standard rules. Arrows qualify, but so do many other kinds of objects. Check closure under both operations first: sums and scalings must stay inside the set. The remaining rules usually ride along from the parent space. **Example.** All polynomials of degree at most 3 form a vector space, and none of them is an arrow. Adding or scaling them never reaches degree 4, and the zero polynomial qualifies. Note the bound says at most: polynomials of degree exactly 3 miss zero and fail. Likewise, symmetric matrices qualify, but invertible matrices do not, since a matrix plus its own negative is the singular zero matrix. A subspace is a slice that is itself a vector space. Demand zero first, then test the two closures: sums of members stay inside, and scalings stay inside. The x-axis passes everything, while the line y equals x plus 1 misses the origin and fails at once. The first quadrant fails scaling, since minus 1 times (1, 1) escapes it. **Tip.** The solutions of a homogeneous system always form a subspace, and linearity is why. If A times x and A times y are both zero, then so are A times their sum and A times any scaling. But solutions of A times x equals b with nonzero b never form a subspace: zero is missing, since A times zero is zero, not b. **Recap.** A vector space closes under addition and scaling, a subspace must also hold zero, and homogeneous solutions always qualify. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Determinants and What They Measure](https://lightmysky.com/learn/mathematics/determinants-and-what-they-measure-mt_fWr2D_1BdM) - [Proof by Deduction and Exhaustion](https://lightmysky.com/learn/mathematics/proof-by-deduction-and-exhaustion-mt_QNWvbkg04f) ## Opens up - [Modules: Linear Algebra Over a Ring](https://lightmysky.com/learn/mathematics/modules-linear-algebra-over-a-ring-mt_0GMVp-s8wD) - [Linear Independence, Span and Basis](https://lightmysky.com/learn/mathematics/linear-independence-span-and-basis-mt_5RGbip9yC9) - [Rings, Fields and Their First Properties](https://lightmysky.com/learn/mathematics/rings-fields-and-their-first-properties-mt_BmhIook13o) - [Nonhomogeneous Equations and Particular Solutions](https://lightmysky.com/learn/mathematics/nonhomogeneous-equations-and-particular-solutions-mt_gbkQZte_JZ) - [The Hahn-Banach Theorem and the Dual Space](https://lightmysky.com/learn/mathematics/the-hahn-banach-theorem-and-the-dual-space-mt_pJuHFxkvjS)