A 2 by 2 matrix can be understood as a rule that moves every point in the coordinate plane to a new point. This makes matrices a powerful way to describe geometric changes such as rotations, stretches, reflections, and shears. Instead of transforming one shape by redrawing it, a matrix transforms the entire grid underneath it.
This viewpoint connects algebra, geometry, computer graphics, physics, and engineering.
Key Facts
- A 2 by 2 matrix A = [[a, b], [c, d]] sends a point (x, y) to (ax + by, cx + dy).
- Matrix transformation formula: [x', y'] = [a x + b y, c x + d y].
- The columns of A show where the basis vectors go: i = (1, 0) maps to (a, c) and j = (0, 1) maps to (b, d).
- A scaling matrix has form [[sx, 0], [0, sy]], so x' = sx x and y' = sy y.
- A rotation by angle theta has matrix [[cos theta, -sin theta], [sin theta, cos theta]].
- The determinant det(A) = ad - bc gives the signed area scale factor of the transformation.
Vocabulary
- Matrix transformation
- A rule that uses a matrix to move each point or vector in the plane to a new location.
- Basis vector
- One of the standard unit vectors i = (1, 0) and j = (0, 1) used to build every vector in the plane.
- Linear transformation
- A transformation that preserves vector addition and scalar multiplication, so grid lines stay straight and parallel lines stay parallel.
- Determinant
- A number from a square matrix that tells how areas scale and whether orientation is preserved or flipped.
- Shear
- A transformation that slants a shape by shifting points in one direction by an amount related to the other coordinate.
Common Mistakes to Avoid
- Multiplying coordinates in the wrong order is wrong because A times a column vector uses rows of the matrix, not columns as separate coordinate multipliers.
- Thinking only the shape moves is wrong because a matrix transformation moves every point in the entire plane, including the grid and axes.
- Assuming every matrix is a rotation is wrong because different entries can create scaling, reflection, shear, collapse, or combinations of these effects.
- Ignoring the determinant is wrong because det(A) tells whether area changes, whether the plane flips orientation, and whether the transformation collapses space when det(A) = 0.
Practice Questions
- 1 Apply the matrix A = [[2, 0], [0, 3]] to the point (4, -1). What are the new coordinates?
- 2 The matrix R = [[0, -1], [1, 0]] represents a 90 degree counterclockwise rotation. Find the image of the points (2, 0), (0, 3), and (1, 1).
- 3 A matrix sends i = (1, 0) to (1, 2) and j = (0, 1) to (3, 1). Explain how you can predict the image of the unit square without transforming every point inside it.