Matrices as Linear Transformations
A matrix is not merely a grid of numbers — it encodes a geometric action on space. Understanding matrices as linear transformations makes matrix multiplication, inverses, determinants, and eigenvalues far more intuitive.
Linear Transformations
A function L that maps vectors to vectors is called a linear transformation if it satisfies two properties for all vectors u, v and all scalars c:
Geometrically: grid lines stay parallel and evenly spaced, and the origin remains fixed. Rotations, reflections, shears, and scalings are all linear transformations. Translations (shifting the origin) are not.
Columns as Images of Basis Vectors
In 2D, any vector (x, y) can be written as x·e₁ + y·e₂ where e₁ = (1,0) and e₂ = (0,1) are the standard basis vectors. By linearity:
This means L is completely determined by knowing L(e₁) and L(e₂). Record these two vectors as the columns of a 2×2 matrix:
where L(e₁) = (a, c) becomes the first column and L(e₂) = (b, d) becomes the second column.
Matrix-Vector Multiplication
Applying A to a vector v = (x, y) means computing x·(first column) + y·(second column):
Written as the standard formula: A·v = [[a,b],[c,d]]·(x,y) = (ax+by, cx+dy). The formula arises naturally from the column interpretation — it is not an arbitrary definition.
Numerical Example
Let A = [[3, 2], [−2, 1]]. What does this transformation do to the vector v = (5, 7)?
- First column (image of e₁): (3, −2). Multiply by 5: (15, −10).
- Second column (image of e₂): (2, 1). Multiply by 7: (14, 7).
- Add: (15+14, −10+7) = (29, −3).
Verify: 3·5+2·7 = 15+14 = 29. −2·5+1·7 = −10+7 = −3. Correct.
Rotation as a Matrix
A 90° counterclockwise rotation maps e₁ = (1,0) to (0,1) and e₂ = (0,1) to (−1, 0). The columns encode these images:
Applying R₉₀ to any vector (x, y) gives (−y, x), which is indeed a 90° counterclockwise rotation.
Frequently Asked Questions
What is a linear transformation?
A function L from vectors to vectors is linear if it preserves addition — L(u+v) = L(u)+L(v) — and scalar multiplication — L(cv) = cL(v). Geometrically, a linear transformation maps straight lines through the origin to straight lines through the origin, keeps the origin fixed, and preserves parallelism and equal spacing.
Why does a 2×2 matrix have exactly four entries?
A 2D linear transformation is completely determined by where it sends the two standard basis vectors e₁=(1,0) and e₂=(0,1). Each image is a 2D vector with two coordinates, giving 2×2 = 4 numbers total.
How do you read a matrix as a transformation?
The first column of the matrix is the image of e₁ and the second column is the image of e₂. Any vector (x,y) maps to x times the first column plus y times the second column.
What does matrix multiplication represent geometrically?
The product BA represents the composition of two linear transformations: first apply A, then apply B. This is why the order matters — BA and AB generally produce different transformations.
Adapted and rewritten from Silas Maths source notes; formulas reviewed for CalxSolver publication.