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, and grid lines stay straight with the origin fixed. Once a basis is fixed, the matrix columns are simply the images of the basis vectors. A 90 degree turn sends (1, 0) to (0, 1) and (0, 1) to (minus 1, 0), giving [[0, minus 1], [1, 0]]. Projection onto the x-axis keeps x and kills y: (3, 5) goes to (3, 0). Reflections work the same way: in the line y = x, (1, 0) lands on (0, 1) and (0, 1) lands on (1, 0), giving columns [[0, 1], [1, 0]].
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, the null space of the matrix. The image is the set of all outputs, the column space. Transposes trade row and null spaces, so keep them handy when you argue about either side.
Dimensions split cleanly: kernel plus image equals domain dimension. For T(x, y) = (x + y, 2x + 2y) the kernel is 1D, so the image is 2 minus 1 = 1D. An onto map from R cubed to R squared has rank 2, leaving kernel dimension 3 minus 2 = 1.
Read a map from its basis images, test it with zero and sums, and split its domain into kernel plus image.
2 · Watch
Take it off screen
Where it sits
8 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.