Linear Transformations Quiz

Study Notes

Reflections

Reflection through the x1-axis: Inverts points across the x1-axis (horizontal axis).
Reflection through the x2-axis: Inverts points across the x2-axis (vertical axis).
Reflection through the line x2 = x1: Swaps the coordinates of points, reflecting across the line that bisects the angle between the axes.
Reflection through the line x2 = -x1: Swaps and negates the coordinates, reflecting across the line at a 45-degree angle in the opposite direction.
Reflection through the origin: Represented by the transformation matrix ([-1, 0; 0, -1]), which inverts points across the origin.

Contraction, Expansion, and Shear

Horizontal contraction and expansion: Changes the x-coordinates of points while keeping the y-coordinates constant; specific transformation matrix not provided.
Vertical contraction and expansion: Changes the y-coordinates of points while preserving the x-coordinates; specific transformation matrix not provided.
Horizontal shear: Moves points horizontally, depending on their y-coordinates; specific transformation matrix not detailed.
Vertical shear: Described by the transformation matrix ([1, 0; k, 1]), modifying the y-coordinates based on x-coordinates, where (k) is the shear factor.

Projections

Projection onto the x1-axis: Projects points onto the x1-axis, effectively setting their x2 coordinates to zero; specific transformation matrix not provided.
Projection onto the x2-axis: Projects points onto the x2-axis, setting their x1 coordinates to zero; specific transformation matrix not provided.

Mappings and Linear Transformations

Onto mapping: A mapping (T: R^n \to R^m) is onto if for every (b) in (R^m), there is at least one (x) in (R^n) such that (T(x) = b).
One-to-one mapping: A mapping (T: R^n \to R^m) is one-to-one if every (b) in (R^m) corresponds to at most one (x) in (R^n).
Conditions for one-to-one transformations: A linear transformation (T: R^n \to R^m) is one-to-one if the equation (T(x) = 0) has only the trivial solution (i.e., (x = 0)).
Conditions involving the standard matrix (A):
- (T) maps (R^n) onto (R^m) if the columns of (A) span (R^m).
- (T) is one-to-one if the columns of (A) are linearly independent.

Description

Test your understanding of linear transformations, including reflections and contractions. This quiz covers key concepts such as transformations across axes and the origin. Get ready to apply these principles to solve problems effectively!