Lecture 15: Introductory Linear Algebra - Linear Transformations and Their Matrices

Introduction

• Topic: Linear transformations and their standard matrices.
• Objective: Explore examples of linear transformations and compute their standard matrices.

Basic Linear Transformations

Zero Transformation

• Definition: Maps every vector to the zero vector.
• Matrix Representation: Zero matrix, with all entries as zero.

Identity Transformation

• Definition: Maps every vector to itself.
• Matrix Representation: Identity matrix, with ones on the diagonal and zeros elsewhere.

Diagonal Transformations

• Definition: Does not mix vector entries; each output entry depends only on one input entry.
• Matrix Representation: Diagonal matrix, with scalar multiples on the diagonal and zeros elsewhere.
• Geometric Interpretation: Stretches along coordinate axes without creating skewed grids.

Projection Transformations

Projection onto the X-axis

• Definition: Maps a vector to its shadow on the x-axis.
• Matrix Representation: \[\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\].

Projection onto a Line

• Specification: Uses a unit vector to define the direction of the line.
• Matrix Representation: \[P_u = u u^T\].
• Example: Given vector w = [2, 1, -2], first normalize w to u and then compute P_u.

Rotation Transformations

90 Degrees Counterclockwise Rotation

• Definition: Rotates vectors by 90 degrees counterclockwise.
• Matrix Representation: \[\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\].

General Rotation by Angle θ

• Matrix Representation:
  • First column: cos(θ), sin(θ)
  • Second column: -sin(θ), cos(θ)
• Example: Rotate vector [1, 3] by 45 degrees using the rotation matrix for θ = π/4.

Conclusion

• Next class: Explore what happens when linear transformations are applied sequentially.

Key Concepts to Remember

• Standard Matrices: Unique matrices representing linear transformations.
• Geometric Interpretation: Understanding how transformations like rotations and projections alter vector spaces.
• Matrix Calculation: Practice with computations involving matrix multiplication and standard matrices.