Linear Transformations
Matrices are not just arrays of numbers. They are functions on space — they take vectors as input and produce new vectors as output. These functions are linear transformations: they rotate, scale, shear, and project geometric space.
What Makes It Linear?
A transformation T is linear if it satisfies two properties for all vectors u, v and scalar c:
Together: T preserves the operations of vector addition and scalar multiplication. Lines stay lines. The origin stays fixed.
Matrix = Transformation
Every linear transformation from ℝⁿ to ℝᵐ can be represented as multiplication by an m×n matrix. Conversely, every matrix defines a linear transformation.
Columns as Basis Images
The columns of matrix A tell you exactly where the standard basis vectors land. Column 1 is where e₁ = [1,0]ᵀ goes. Column 2 is where e₂ = [0,1]ᵀ goes. Any other vector is just a linear combination of these images.
Rotation by Angle θ
Rotating every vector in ℝ² counterclockwise by angle θ is a linear transformation. Its matrix:
Common Transformations
- Scaling — diagonal matrix; stretches/compresses axes independently
- Reflection — across x-axis: [[1,0],[0,−1]]; across y-axis: [[−1,0],[0,1]]
- Shear — [[1,k],[0,1]]; slants the plane horizontally
- Projection — onto x-axis: [[1,0],[0,0]]; collapses y-component
- Identity — [[1,0],[0,1]]; maps every vector to itself
Chaining Transformations
Applying transformation A then transformation B is the same as multiplying their matrices: (BA)x = B(Ax). Matrix multiplication IS function composition. Order matters — BA ≠ AB in general. The rightmost matrix is applied first.
Invertible Transformations
A transformation is invertible if it can be undone — if for every output vector there is exactly one input that produces it. Geometrically: the transformation doesn't collapse any dimension. Algebraically: det(A) ≠ 0.
Determinant = Area Scale
|det(A)| equals the factor by which the transformation scales areas (in 2D) or volumes (in 3D). If det = 2, every region doubles in area. If det = 0, all of space collapses into a lower-dimensional subspace — the transformation is not invertible.
Key Takeaways
- Every matrix is a linear transformation; every linear transformation has a matrix
- Columns of A = images of standard basis vectors under the transformation
- Rotation, scaling, shear, projection, reflection — all encoded as matrices
- Matrix multiplication = composition of transformations (apply right-to-left)
- Invertible ↔ det(A) ≠ 0 ↔ transformation doesn't collapse space
Module 3: Systems of Equations
With matrices as transformations, solving Ax = b becomes: find the input vector x whose image under A is b. This geometric view transforms elimination into something elegant — and sets the stage for row reduction, rank, and null space.