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.
What a learner can do afterwards
- 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
1 · Read
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]].
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.
Read a map from its basis images, test it with zero and sums, and split its domain into kernel plus image.
Reading summaries adapted from Khan Academy articles on linear transformations. The kernel-and-image reading is original. Videos by Khan Academy. Practice questions are original. The rewritten matrix reading was written for this dataset.
2 · Watch
Take it off screen
Where it sits
14 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.