--- title: "Linear Independence, Span and Basis" description: "Span is everything you can reach by combining vectors; independence says none of them is redundant. A basis is a set that is both, so every element has exactly one expression." canonical: https://lightmysky.com/learn/mathematics/linear-independence-span-and-basis-mt_5RGbip9yC9 source: https://lightmysky.com/learn/mathematics/linear-independence-span-and-basis-mt_5RGbip9yC9.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`. # Linear Independence, Span and Basis Span is everything you can reach by combining vectors; independence says none of them is redundant. A basis is a set that is both, so every element has exactly one expression. Subject: Mathematics · Area: Linear Algebra · Ages 18 to 19 Page: https://lightmysky.com/learn/mathematics/linear-independence-span-and-basis-mt_5RGbip9yC9 ## Ready when they can - Decide independence by solving a homogeneous system - Describe the span of a set of vectors geometrically - Show that coordinates with respect to a basis are unique ## Lesson: Reaching everywhere with nothing spare Vectors are independent when only the trivial combination gives zero: if scaled copies add to zero, every scale must be zero. Test this by solving the homogeneous system. A shortcut for pairs: if one vector is a multiple of the other, the pair is dependent. Merely avoiding the zero vector is not enough. The span of a set is everything you can build by combining its members. One nonzero vector spans a line through the origin in its direction. Two vectors that are not multiples span a whole plane. Since two already span the plane, any third vector in it depends on them, and three is the minimum that can span all of three dimensional space. **Example.** A basis is an independent set that spans the whole space. The pair (1, 0) and (0, 1) is a basis of the plane. The vector (5, 7) is 5 times (1, 0) plus 7 times (0, 1), and independence makes this the only way. Three vectors in the plane can never form a basis: one of them always depends on the rest. **Tip.** To show coordinates are unique, suppose two writings agree and subtract them. The difference is a combination giving zero, so independence forces every coefficient to match. That is why each vector has exactly one expression in a basis. Remember the chain: solve the system, read the geometry, then trust the basis. **Recap.** Independence means no redundancy, span means full reach, and a basis gives both, so coordinates are unique. ## Practice 16 questions on this page, each with its working shown. ## Needs first - [Gaussian Elimination and Row Echelon Form](https://lightmysky.com/learn/mathematics/gaussian-elimination-and-row-echelon-form-mt_PZBce3UvRm) - [Vector Spaces and Subspaces](https://lightmysky.com/learn/mathematics/vector-spaces-and-subspaces-mt_Zrddx-E6_n) ## Opens up - [Dimension and the Rank-Nullity Theorem](https://lightmysky.com/learn/mathematics/dimension-and-the-rank-nullity-theorem-mt_aMG8PcwPAR) - [Field Extensions and Their Degrees](https://lightmysky.com/learn/mathematics/field-extensions-and-their-degrees-mt_CjOMNt5_i0) - [The Gram-Schmidt Process](https://lightmysky.com/learn/mathematics/the-gram-schmidt-process-mt_nkp231EA_E)