[Linear Algebra] 7. Nonsquare matrices as transformations between dimensions

In this section, let's think about non-square matrices geometrically.

 

Non-square matrics are transformations that transform particular dimensional vectors into other dimensional vectors.

2-D (plane) to 3-D (space)

If there is a transformation that takes $\mathbf{\hat{i}}$ to the coordinate $\begin{bmatrix} a \\ b \\ c \end{bmatrix}$ and $\mathbf{\hat{j}}$ to the coordinate $\begin{bmatrix} d \\ e \\ f \end{bmatrix}$, we can describe this transformation by $3 \times 2$ matrix.

 

The $2$ columns indicate that the input space has two basis vectors and the $3$ rows indicate that the landing spot of each basis vector are described with $3$ coordinates. So we can interpret this transformation as 2-D to 3-D transformation.

 

The column space of matrix, place where all the vectors land, is a 2-D plane slicing through the origin of 3-D space. So the matrix is full rank. Therefore, we can interpret this transformation as 2-D to 2-D transformation.

 

In a nutshell, we can interpret this as "Although the form of output vector is 3-D, like $\begin{bmatrix} a \\ b \\ c \end{bmatrix}$, it only span 2-D". Surprisingly, applying another linear transformation doesn't span a dimension larger than two dimensions.

3-D (space) to 2-D (plane)

This linear transformation can be described by $2 \times 3$ matrix.

 

The $3$ columns indicate that the input space has three basis vectors and the $2$ rows indicate that the landing spot of each basis vector are described with $2$ coordinates. So we can interpret this transformation as 3-D to 2-D transformation.

 

The column space of matrix is a 2-D plane slicing through the origin of 3-D space. So we can interpret this transformation as 3-D to 2-D transformation.

 

In a nutshell, we can interpret this as "the form of output vector and column space is 2-D".

 

But also we can interpret this transformation as 3-D to 3-D transformation. By thinking, this as a projection.

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Why there is a difference between those transformations? Because transformation is a function. 1). Dimension expansion is similar to one-to-many relation, which is not function. But 2). dimension reduction is similar to many-to-one relation, which is function.