Linear transformations and matrices | Chapter 3, Essence of linear algebra

Linear Transformations and Matrices

Introduction to Linear Transformations

• Linear Transformation: A function that takes in a vector and outputs another vector, visualized in terms of movement.

• Function vs. Transformation: The term transformation suggests a visual aspect of the input-output relation where vectors move to their corresponding outputs.

Visualization of Transformations in 2D

• Movement Representation: To grasp transformations, visualize each vector as a point in space moving to another point.

• Infinite Grid: Using an infinite grid helps track the transformation effect on all points, showing how space is modified beautifully.

Properties of Linear Transformations

• Linearity Conditions:

  • Straight Lines: All straight lines remain straight without distortion.

  • Origin Fixation: The origin (0,0) must remain fixed.

• Examples:

  • A transformation that curves lines or moves the origin is not linear.

  • Visual understanding requires analyzing effects on both horizontal and diagonal lines.

Understanding Transformations Through Basis Vectors

• Basis Vectors: The unit vectors i-hat (x-axis) and j-hat (y-axis) determine how other vectors are transformed.

• Transformation Impact: Following where i-hat and j-hat land informs how any vector transforms:

  • Given a vector v = (-1, 2):

    • If i-hat lands at (1, -2) and j-hat lands at (3, 0), then

      • v transforms as:

        • v' = -1 * (1, -2) + 2 * (3, 0) = (5, 2).

Formulating Linear Transformations Mathematically

• Coordinates Representation: Each transformation can be completely defined by the end coordinates of the basis vectors.

• Matrix Representation:

  • A 2x2 matrix represents these transformations:

    • First column for i-hat's landing coordinates.

    • Second column for j-hat's landing coordinates.

• General Matrix Structure:

  • If matrix = [[A, B], [C, D]], the transformation can be applied to a vector (x,y) resulting in:

    • (Ax + By, Cx + Dy).

Practical Examples of Transformations

• Rotation Example: 90-degree counterclockwise rotation:

  • i-hat -> (0, 1)

  • j-hat -> (-1, 0)

  • Resulting matrix: [[0, -1], [1, 0]]

• Shear Example:

  • i-hat remains at (1, 0) while j-hat moves to (1, 1).

  • Resulting matrix: [[1, 1], [0, 1]]

Deducing Transformations from Matrices

• Identifying Transformation: Given a matrix with specific columns, determine how i-hat and j-hat move to visualize transformation.

• Linear Dependence: If columns of the matrix are linearly dependent, all of space collapses along a line.

Key Takeaways

• Understanding Linear Transformations: Keep the definitions and properties in mind. They move space while preserving structure.

• Matrix as a Language: Treat matrices as descriptions of linear transformations; they encode vital information in their columns.

Importance in Linear Algebra

• Foundation for Topics: The understanding of transformations sets the stage for advanced concepts including matrix multiplication, determinants, eigenvalues, etc.

• Conclusion: A strong grasp of these ideas simplifies further study and appreciation of linear algebra's depth.