--- title: "Linear Maps and Their Matrices" description: "A linear map respects addition and scaling. Once a basis is fixed, every such map is a matrix, and the matrix columns are the images of the basis vectors." canonical: https://lightmysky.com/learn/mathematics/linear-maps-and-their-matrices-mt_pzrfiQFdto source: https://lightmysky.com/learn/mathematics/linear-maps-and-their-matrices-mt_pzrfiQFdto.md retrieved: 2026-11-12 --- > **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 Maps and Their Matrices A linear map respects addition and scaling. Once a basis is fixed, every such map is a matrix, and the matrix columns are the images of the basis vectors. Subject: Mathematics · Area: Linear Algebra · Ages 19 to 20 Page: https://lightmysky.com/learn/mathematics/linear-maps-and-their-matrices-mt_pzrfiQFdto ## Ready when they can - Build the matrix of a rotation or projection from images of the basis vectors - Check linearity from the two defining properties - Identify the kernel and image of a map and relate them to null space and column space ## Lesson: Matrices are maps on basis vectors A linear map respects addition and scaling: grid lines stay straight and the origin stays fixed. Once a basis is fixed, the matrix columns are the images of the basis vectors. A quarter turn clockwise sends (1, 0) to (0, minus 1) and (0, 1) to (1, 0), giving [[0, 1], [minus 1, 0]]. Projection onto the y-axis keeps y and kills x: (4, 2) goes to (0, 2). Reflection in the x-axis keeps (1, 0) and sends (0, 1) to (0, minus 1), giving [[1, 0], [0, minus 1]]. *(drawing: Each basis image becomes one column of the matrix.)* **Example.** Test linearity fast. Every linear map sends zero to zero, so a map sending zero to (1, 0) already fails. The two rules to check are T(u + v) = T(u) + T(v) and T(c v) = c T(v). Draw a small shape, move the basis vectors, and confirm the shape follows. The kernel is the set of vectors mapping to zero: for the matrix of the map it is the null space, found by solving A x = 0. The image is the set of all outputs: the column space, spanned by the matrix columns. For T(x, y) = (x minus y, 3x minus 3y), the kernel is the line y = x, and every output is a multiple of (1, 3), so the image is the line through (1, 3). Dimensions split cleanly: kernel dimension plus image dimension equals the domain dimension, which is rank plus nullity for the matrix. For T(x, y) = (x minus y, 3x minus 3y) the kernel is 1D, so the image is 2 minus 1 = 1D. An onto map from four dimensional space to R squared has rank 2, leaving kernel dimension 4 minus 2 = 2. **Recap.** Read a map from its basis images, test it with zero and sums, and split its domain into kernel plus image. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Dimension and the Rank-Nullity Theorem](https://lightmysky.com/learn/mathematics/dimension-and-the-rank-nullity-theorem-mt_aMG8PcwPAR) ## Optional connections - [Function notation and evaluating functions](https://lightmysky.com/learn/mathematics/function-notation-and-evaluating-functions-mt_YWKWzHK_6Z) ## Opens up - [Groups, Subgroups and Symmetry](https://lightmysky.com/learn/mathematics/groups-subgroups-and-symmetry-mt_HFAhcaGo2L) - [Change of Basis](https://lightmysky.com/learn/mathematics/change-of-basis-mt_HTGuJhX484) - [Bounded Linear Operators and the Operator Norm](https://lightmysky.com/learn/mathematics/bounded-linear-operators-and-the-operator-norm-mt_Qi_mbSNMMx) - [Group Representations and the Group Algebra](https://lightmysky.com/learn/mathematics/group-representations-and-the-group-algebra-mt_y5JlvQGKYW)