Non-square Matrices as Transformations Between Dimensions

So far we have only covered two-dimensional vectors to other two-dimensional vectors, represented by 2 x 2 matrices. Or three-dimensional vectors to other three dimensional vectors, represented by 3 x 3 matrices.

Non-Square Matrices

By now we have most of the tools that we need to be able to ponder on non-square matrices.

  i  j
[ 0 -2 ]
[ 5  1 ]
[ 1  4 ]

It is perfectly reasonable to talk about transformations between dimensions, which take two-dimensional vectors and transform them to three-dimensional vectors.

[ 2 ]          [ 1 ]
[ 7 ]-> L(v) ->[ 8 ]
               [ 2 ]

Again, what makes these transformations linear is that the gridlines remain parallel and evenly spaced. Along with the fact that the origin remains fixed in space.

It is important to understand that two-dimensional inputs are very different from their corresponding three-dimensional outputs.

Describing the Transformation

Encoding one of these transformations is the same as what we have done before. You take a look at where the basis vectors lands and write the coordinates of each landing spot as the columns of a matrix. The following matrix takes i to the coordinates of (2, -1, -2) and j to the coordinates of (0, 1, 1).

  i  j
[ 2  0 ]
[-1  1 ]
[-2  1 ]

This means that the matrix used to encode our transformation has 3 rows and 2 columns. Meaning that we are using a 3 x 2 matrix.

The Columns Space

In the language of the last topic, the column space of this matrix, the place where all the vectors land, is a two-dimensional plane slicing through the origin of three-dimensional space. But the matrix is still full rank, since the number of dimensions in the column space is the same as the number of dimensions in the input space (rank of 2 with only 2 columns).

The Relationship Between Columns and Rows

So if you see a 3 x 2 matrix, you can know that it has the geometric interpretation of mapping two-dimensions to three-dimensions.

[ 1  1 ]
[ 0  1 ]
[ 1  0 ]

Since we know that the 2 columns indicate that the input space has only two basis vectors, meaning that we are starting in two-dimensions. While the three rows indicate that the landing spot for each of those basis vectors is described with three separate coordinates, meaning that they are landing in three-dimensions.

Likewise, if you see a 2 x 3 matrix with two rows and three columns, it indicates that the input space has three basis vectors, meaning that you are starting in three-dimensions. While the landing spot for each of those basis vectors is described with only two separate coordinates, meaning that they are landing in two dimensions.

[ 3  1  4 ]
[ 1  5  9 ]

From Two-Dimensions to a Number Line

You can also have a transformation from two dimensions to one dimension. One dimensional space is really just the number line. Therefore a transformation like this takes two-dimensional vectors, and spits out numbers.

Thinking of the concept of gridlines remaining parallel and evenly spaced can get a little messy here. Therefore the understanding for what linearity means is that if you have a line of evenly spaced dots, they would remain evenly spaced after they’re mapped onto the number line.

One of these transformations is encoded by a 1 x 2 matrix. Each of whose columns are just a single entry. Each of those columns represents where those basis vectors land, and since we are on a flat number line, each of those columns only requires one number, number where that basis vectors lands on (see picture one).

[ 1  2 ]

This is a surprising meaningful transformation, which is closely related to the dot product, which we will cover in our next post.