Linear Transformations Quiz

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.

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.

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.

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.