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• Hey, everyone!

• I've got another quick footnote for you between chapters today.

• When I talked about linear transformation so far,

• I've only really talked about transformations from 2-D vectors to other 2-D vectors,

• represented with 2-by-2 matrices;

• or from 3-D vectors to other 3-D vectors, represented with 3-by-3 matrices.

• But several commenters have asked about non-square matrices,

• so I thought I'd take a moment to just show with those means geometrically.

• By now in the series, you actually have most of the background you need

• to start pondering a question like this on your own.

• But I'll start talking through it, just to give a little mental momentum.

• It's perfectly reasonable to talk about transformations between dimensions,

• such as one that takes 2-D vectors to 3-D vectors.

• Again, what makes one of these linear

• is that grid lines remain parallel and evenly spaced, and that the origin maps to the origin.

• What I have pictured here is the input space on the left, which is just 2-D space,

• and the output of the transformation shown on the right.

• The reason I'm not showing the inputs move over to the outputs, like I usually do,

• is not just animation laziness.

• It's worth emphasizing the 2-D vector inputs are very different animals from these 3-D

• vector outputs,

• living in a completely separate unconnected space.

• Encoding one of these transformations with a matrix is really just the same thing as

• what we've done before.

• You look at where each basis vector lands

• and write the coordinates of the landing spots as the columns of a matrix.

• For example, what you're looking at here is an output of a transformation

• that takes i-hat to the coordinates (2, -1, -2) and j-hat to the coordinates (0, 1, 1).

• Notice, this means the matrix encoding our transformation has 3 rows and 2 columns,

• which, to use standard terminology, makes it a 3-by-2 matrix.

• In the language of last video, the column space of this matrix,

• the place where all the vectors land is a 2-D plane slicing through the origin of 3-D

• space.

• But the matrix is still full rank,

• since the number of dimensions in this column space is the same as the number of dimensions

• of the input space.

• So, if you see a 3-by-2 matrix out in the wild,

• you can know that it has the geometric interpretation of mapping two dimensions to three dimensions,

• Since the two columns indicate that the input space has two basis vectors,

• and the three rows indicate that the landing spots for each of those basis vectors

• is described with three separate coordinates.

• Likewise, if you see a 2-by-3 matrix with two rows and three columns, what do you think

• that means?

• Well, the three columns indicate that you're starting in a space that has three basis vectors,

• so we're starting in three dimensions;

• and the two rows indicate that the landing spot for each of those three basis vectors

• is described with only two coordinates,

• so they must be landing in two dimensions.

• So it's a transformation from 3-D space onto the 2-D plane.

• A transformation that should feel very uncomfortable if you imagine going through it.

• You could also have a transformation from two dimensions to one dimension.

• One-dimensional space is really just the number line,

• so transformation like this takes in 2-D vectors and spits out numbers.

• Thinking about gridlines remaining parallel and evenly spaced

• is a little bit messy to all of the squishification happening here.

• So in this case, the visual understanding for what linearity means is that

• if you have a line of evenly spaced dots,

• it would remain evenly spaced once they're mapped onto the number line.

• One of these transformations is encoded with a 1-by-2 matrix,

• each of whose two columns as just a single entry.

• The two columns represent where the basis vectors land

• and each one of those columns requires just one number, the number that that basis vector

• landed on.

• This is actually a surprisingly meaningful type of transformation with close ties to

• the dot product,

• and I'll be talking about that next video.

• Until then, I encourage you to play around with this idea on your own,

• contemplating the meanings of things like matrix multiplication and linear systems of

• equations

• in the context of transformations between different dimensions.

• Have fun!

Hey, everyone!

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# Nonsquare matrices as transformations between dimensions | Essence of linear algebra, chapter 8

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tai posted on 2021/02/07
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