Determinants and Linear Transformations

Linear Transformations

• Visual Understanding: Linear transformations can stretch or squish space.
• Matrix Representation: Each transformation is represented by a specific matrix.

Measuring Area Change

• Stretching/Squishing: Assess transformations by how much they scale an area's size.
• Example:
  • Matrix: [3, 0; 0, 2]
  • Effect: Scales i-hat by 3 and j-hat by 2.
  • 1x1 Square to 2x3 Rectangle: Area scaled by factor 6.

Shear Transformation

• Matrix: [1, 0; 1, 1]
• Effect: i-hat stays, j-hat moves to [1, 1].
• Area Change: Unit square turns into a parallelogram with unchanged area.

General Scaling Factor

• Implication: Area changes uniformly across space.
• Special Factor: Called the determinant.
• Determinant Examples:
  • 3: Increases area 3 times.
  • ½: Decreases area half.
  • 0: Squishes space to line/point (0 area).

Identification via Determinant

• Zero Determinant: Indicates transformation squishes space into a smaller dimension.
• Utility: Provides insight into whether transformation decreases dimension.

Determinant and Orientation

• Negative Determinants:
  • Implication: Indicates flipping of orientation in space.
  • Absolute Value: Shows the scaling factor.
  • Example: Matrix [1, 1; 2, -1] has determinant -3.
• Orientation Flip: Check using i-hat and j-hat positions.

3D Determinants

• Volume Change: Measures how transformations affect volume.
• Focus on Unit Cube: [i-hat, j-hat, k-hat].
• Shape After Transformation: Parallelipiped.
• Determinant of 0: Indicates volume becomes 0 (space squished).
  • Linear Dependence: Functions involving linearly dependent columns.
• Right-Hand Rule: Used to assess orientation change.

Computing the Determinant

• 2x2 Matrix: det(A) = ad - bc for matrix [a, b; c, d].
• Intuition:
  • If b and c are 0, i-hat stretched by a and j-hat by d.
  • Non-zero b, c: Parallelogram's area approximations.
• Practice: Essential for computing longer matrices.

Further Thoughts

• Matrix Multiplication: Determinant of product matrix equals product of individual determinants.
• Next Topic: Relating linear transformations to linear systems of equations.