Linear Algebra · Lesson 9
Linear Transformations
A linear transformation T maps vectors to vectors while preserving addition and scalar multiplication. Every linear transformation can be represented by a matrix — and every matrix represents a linear transformation. This is the deep heart of linear algebra.
Quick Check
1. The rotation matrix for 90° CCW is:
[[1,0],[0,1]]
[[0,−1],[1,0]]
[[0,1],[−1,0]]
[[−1,0],[0,−1]]
2. Which matrix projects every vector onto the x-axis?
[[1,0],[0,1]]
[[1,0],[0,0]]
[[0,0],[0,1]]
[[1,1],[0,0]]
3. To find the matrix of T, you need to know:
The determinant of T
Where T sends each standard basis vector
The inverse of T
The eigenvalues of T
4. If T reflects over the y-axis, then T(3, −2) = ?
(3, 2)
(−3, −2)
(−2, 3)
(2, −3)