Linear transformations don’t always have to stay within the same number of dimensions. They can map from one space (like 2D) to a completely different space (like 3D), or even from 3D to 1D. Here’s a simple breakdown of the key ideas, based on the concepts discussed:
Key Ideas About Non-Square Matrices
- 3x2 Matrix (2D to 3D):
- Maps 2D space into 3D space.
- The outputs lie on a plane in the 3D space defined by the two columns.
- 2x3 Matrix (3D to 2D):
- Maps 3D space into 2D space.
- The outputs are a flattened projection of the 3D space onto a 2D plane.
- 1x2 Matrix (2D to 1D):
- Maps 2D space into 1D (the number line).
- The result is a single number, often related to a projection or a measurement.
Why Does This Matter?
- Non-square matrices let us understand transformations between dimensions, not just within the same dimension.
- The number of rows tells you the dimensions of the output space.
- The number of columns tells you the dimensions of the input space.
- The columns of the matrix determine where the basis vectors of the input space land, and everything else follows from that.
Takeaway:
- 3x2 Matrix: Takes 2D inputs and maps them into a 3D plane.
- 2x3 Matrix: Projects 3D inputs into a flat 2D space.
- 1x2 Matrix: Compresses 2D inputs down to a single number.
Understanding how matrices with different dimensions affect space helps in grasping transformations like projections, scaling, and dimension reduction!