Linear Transformations in ℝ³ and 3×3 Matrices

Linear Transformations in ℝ³ and 3×3 Matrices

Matrices give us a compact way to describe linear transformations in three-dimensional space. A linear transformation is a mapping T : ℝ³ → ℝ³ that sends a point with position vector (x, y, z) to another point, according to a rule with two key properties.

What Makes a Transformation Linear?

A transformation T : ℝ³ → ℝ³ is called linear if, for all real numbers λ and all vectors (x, y, z) in ℝ³,

T(λx, λy, λz) = λ T(x, y, z),

and for all vectors (x₁, y₁, z₁) and (x₂, y₂, z₂) in ℝ³,

T(x₁ + x₂, y₁ + y₂, z₁ + z₂) = T(x₁, y₁, z₁) + T(x₂, y₂, z₂).

The point that (x, y, z) is sent to is called the image of (x, y, z) under T.

The Standard Basis Vectors

To find the matrix that represents a particular transformation, it is enough to know what happens to three special vectors, called the standard basis for ℝ³:

  • î = (1, 0, 0)
  • ĵ = (0, 1, 0)
  • k̂ = (0, 0, 1)

Once we know the images of î, ĵ and k̂, the entire matrix for T is determined. Each image becomes one column of the matrix that represents T.

From a Matrix to Its Action on ℝ³

Let M be any 3×3 matrix. We can describe it in terms of its columns:

M has columns (a, d, g), (b, e, h), (c, f, i).

When M is used as a linear transformation on ℝ³, we obtain:

  • M(1, 0, 0) = (a, d, g)
  • M(0, 1, 0) = (b, e, h)
  • M(0, 0, 1) = (c, f, i)

So the image of (1, 0, 0) is (a, d, g), the image of (0, 1, 0) is (b, e, h), and the image of (0, 0, 1) is (c, f, i). This shows that the columns of M are exactly the images of the standard basis vectors.

Combining Transformations with Matrix Multiplication

Suppose the transformation T is represented by a matrix, also written T, and another transformation U is represented by a matrix U. The transformation “apply T, then apply U” is written U ∘ T and is represented by the matrix product UT.

U ∘ T ←→ UT

In words: first transform the vector using T, then transform the result using U. Matrix multiplication encodes this composition of transformations.

Reading the Notation T : ℝ³ → ℝ³

The symbol ℝ³ denotes the set of all 3D position vectors. Writing T : ℝ³ → ℝ³ is read as:

“The transformation T maps points in three-dimensional space to points in three-dimensional space.”

These ideas form a foundation for working with rotations, reflections, scalings, and many other geometric transformations using 3×3 matrices.

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