Dimensions made easy

I could have a matrix like this-- 3, 7, and Well, this has one row.

This is the one row that we see here. This is a 1 by 3 matrix.

I could have a matrix-- and I think you see where all of this is going. Figuring out the dimensions of a matrix are not too difficult. I could have a matrix that looks like this, where it's 3, 5, 0, 0, negative 1, negative 7. This right over here has three rows. So it's three rows, and it has two columns. So we would call this a 3 by 2. Let me do that in that same color. We would call it a 3 by 2 matrix, three rows and two columns. You know that a matrix is just a rectangular array of numbers.

You can say what its dimensions are. You know that each of these numbers that take one of these positions-- we just call those entries.

But what are matrices good for? I still might not be clear what the connection is between this and this right over here. And at the most fundamental level, this is just a compact representation of a bunch of numbers. It's a way of representing information.

First Dimension

Now you're saying, "OK, that's cute and everything, "but this seems like a little bit of too much overhead "to worry about when I'm just doing "a simple formula like this. It's useful for something as simple as distance equals rate times time, but as you go into physics and chemistry and engineering, you'll see much, much, much more, I would say, hairy formulas.

When you do the dimensional analysis, it makes sure that the math is working out right. It makes sure that you're getting the right units.

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But even with this, let's try a slightly more complicated example. Let's say that our rate is, let's say, let's keep our rate at 5 meters per second, but let's say that someone gave us the time.

Instead of giving it in seconds, they give it in hours, so they say the time is equal to 1 hour. Now let's try to apply this formula. We're going to get distance is equal to 5 meters per second, 5 meters per second times time, which is 1 hour, times 1 hour. What's that going to give us? The 5 times the 1, so we multiply the 5 times the 1, that's just going to give us 5.

Introduction to matrices

But then remember, we have to treat the units algebraically. We're going to do our dimensional analysis, so it's 5, so we have meters per second times hours, times hours, or you could say 5 meter hours per second. Well, this doesn't look like a This isn't a set of units that we know that makes sense to us. This doesn't feel like our traditional units of distance, so we want to cancel this out in some way. It might jump out of you, well, if we can get rid of this hours, if we can express it in terms of seconds, then that would cancel here, and we'd be left with just the meters, which is a unit of distance that we're familiar with.

So how do we do that? We'd want to multiply this thing by something that has hours in the denominator and seconds in the numerator, times essentially seconds per hour. Lines all have the same width and the same height, but their length can be varied. If you want to make lines varying thicknesses, then you have to move into 2-dimensional lines.

A cube in the first dimension would look like a line with the same length as said cube, but no width or height values. Transforming a line segment in a direction perpendicular to the 1-dimensional direction brings you into the second dimension. Keep this idea in mind as we expand our dimensional knowledge and notice this perpendicular action repeated as we move through dimensions. In the second dimension, our cube can begin to look like a cube, but only just barely.

A cube would exist as a square in the second dimension. You can, of course, draw a representation of a 3D cube in 2 dimensions, but that isn't what a cube would look like in 2 dimensions. Rather this would simply be a representation of the third dimension superimposed on the second. Length and width can be varied in the second dimension, which allows for basic shapes and geometry.

When we move into the third dimension, the math starts getting more complex. The cube from the second dimension now gets extruded in a third perpendicular direction to both sides of the 2D square. To put this in cartesian terms, the 2D square existed in the X and Y directions. Moving into the 3rd dimension extruded that square in the Z direction. The third dimension is where our cube actually becomes a cube in our traditional defined sense. The object has dimensions of width, length, and height. Throughout all of the dimensions, it is important to note that a cube will maintain all of its basic properties in theory.

All of the angles will be right and all of the sides will be the same. Bringing in another principle of dimensions, we can examine what would occur if the cube was expanded indefinitely.