Linear Transformations

You’ll be able to

Interpret a 2×2 matrix as a transformation of the plane
Identify the matrices for common reflections
Explain the effect of the identity matrix on a vector

Matrices move the plane

Multiplying a 2×2 matrix by a column vector produces a new vector — the matrix acts as a linear transformation, sending each point of the plane to a new location. Rotations, reflections, stretches, and shears can all be encoded as a single matrix. The columns of the matrix tell you where the basis vectors ⟨1, 0⟩ and ⟨0, 1⟩ land.

Reflections and the identity

A reflection across the x-axis keeps x and negates y, encoded by [[1, 0], [0, −1]]. A reflection across the y-axis negates x instead: [[−1, 0], [0, 1]]. The identity matrix [[1, 0], [0, 1]] changes nothing — multiplying it by any vector returns that same vector, just as multiplying a number by 1 leaves it unchanged.

Common 2×2 transformations

reflect x-axis [[1, 0], [0, −1]] · reflect y-axis [[−1, 0], [0, 1]] · identity [[1, 0], [0, 1]]

Each column shows where a basis vector goes. The identity leaves every vector fixed; a reflection flips the sign of one coordinate.

Worked example

Apply the matrix [[1, 0], [0, −1]] to the vector ⟨3, 5⟩ and describe the transformation.

1.Multiply: the new x is (1)(3) + (0)(5) = 3.
2.The new y is (0)(3) + (−1)(5) = −5.
3.The image is ⟨3, −5⟩ — the x-coordinate is unchanged and the y-coordinate is negated.
4.Negating only y is a reflection across the x-axis.

Answer: The result is ⟨3, −5⟩, a reflection across the x-axis: x stays the same while y flips sign.

Checkpoint

Which matrix reflects a point across the x-axis (keeping x, negating y)?

Tip

Read a transformation matrix by its columns: the first column is the image of ⟨1, 0⟩ and the second is the image of ⟨0, 1⟩. Knowing where the two basis vectors go pins down the whole transformation.

Checkpoint

Applying the 2×2 identity matrix [[1, 0], [0, 1]] to a vector produces:

On the exam

To find the matrix of a described transformation, track where ⟨1, 0⟩ and ⟨0, 1⟩ land and place those images as columns. This construction handles rotations, reflections, and stretches uniformly on the AP exam.

Answer the 2 checkpoints as you read.