March 30, 2008 7:33 PM Subscribe

I tutor several people in a "Mathematics for the Social Sciences" course at a local university. This semester the material is focused on matrix algebra. For the most part, they have no trouble wrapping their heads around the notions of basic vector operations, row reduction, systems of equations, inverse matrices and determinants. But, they have a lot of problems with more abstract concepts such as linear independence, subspaces, basis and dimension. I'm wondering if anyone has any tips on helping them visualize and understand these ideas more clearly.

posted by Proginoskes to Education (7 answers total) 8 users marked this as a favorite

posted by Proginoskes to Education (7 answers total) 8 users marked this as a favorite

Demonstrate the abstract concepts using metaphors from computer graphics.

posted by Blazecock Pileon at 7:44 PM on March 30, 2008

posted by Blazecock Pileon at 7:44 PM on March 30, 2008

I'd say pictures help. vectors and a 3d cartesian plot. 2 vectors on the X-Y plane clearly cannot cover all of 3D space, but they can form a 2d subspace, etc etc.

posted by Large Marge at 8:20 PM on March 30, 2008

posted by Large Marge at 8:20 PM on March 30, 2008

You might try a little dip into color spaces and color theory (at the very shallow oversimplified end). They might intuitively believe that if you show them a color and ask them to mix red, yellow, and blue paints to match the color, that there will be just one correct recipe. And obviously if, instead of red, you gave them some other paint that was already a mixture of blue and yellow, well, that wouldn't do shit for them.

Of course this won't hold up if you go too far, but the point is to get away from the arrows and axes and mindless grinding out of matrix operations. Treat the coordinates as amounts of some concrete property and think up some goofy examples. Determinants are more abstract than basis and dimension!

posted by fleacircus at 1:31 AM on March 31, 2008

Of course this won't hold up if you go too far, but the point is to get away from the arrows and axes and mindless grinding out of matrix operations. Treat the coordinates as amounts of some concrete property and think up some goofy examples. Determinants are more abstract than basis and dimension!

posted by fleacircus at 1:31 AM on March 31, 2008

This site is great for learning to visualize concepts in linear algebra. The only downside is that it uses somewhat non-standard terminology.

posted by dfan at 7:27 AM on March 31, 2008 [1 favorite]

posted by dfan at 7:27 AM on March 31, 2008 [1 favorite]

The most important (constructive) concept I think is the idea of basis elements. Once people get them intuitively at a gut level, criteria for bases (linear independence) can be formalized, and the importance is manifest. But the basis lets you describe your space (physical space, state space, personality space, analytical space, etc....), so if you're going to be talking about STUFF, you gotta understand your basis.

I'm with wanderingmind: 1D is boring, 2D is interesting and non-trivial, so I'd start there. Later 2D can be described as a subspace of 3D. The idea is to build people's intuition with something that is a real honest linear space, so I'd stay away from color mixing for a bit. Let people navigate physical space, which is what they've been doing all their lives, then you can give them other things to think about later. Here's a sketch off the top of my head of how I'd play with this given a TA section or 2.

First, orthogonal:

Everyone can navigate Manhattan, given directions, yeah? There are two nice basis vectors (North and East), that I think everyone gets. People can see that there's no way you can describe east in terms of north (without rotation), and vice versa, but with both you can describe every cardinal direction, in a way you couldn't if all you had was a "West arrow" and an "East arrow". There's loads of examples in Real City Life you can use to discuss normalization, superposition, all applied linear algebra fun. So you can play with different representations using orthogonal 2-d basis.

Then, linear independence:

Can you get everywhere if someone gives you NEE and East? Sure, all you need is a little of the north, and you can still get everywhere. Important bit is the spanning of the space.

But what about getting to the 21st floor? We need a new linearly independent vector to get to 3D (2D was a subspace etc). If these were entry-physicists, this would be the time to introduce the Minkowski metric, and Time for 4D, social scientists should know this too, but you might have a rebellion on your hands.

So after people have this language for talking about physical space, let them relax and encourage them to start thinking about everything in terms of spanning basis (personality === Cambell's archetypes. taste === salty, sweet, etc. fashion, music, ..., everything you can describe you can whip up some sort of basis for, although for some things (functions of real variables, eg.) you need infinite bases). Some of these things don't really encourage the analogy of linear superposition, some do, and it's valuable for scientists (even the social kind) to learn the boundaries of the language their using.

Start with the physical though.

posted by johnjoe at 12:11 PM on March 31, 2008 [2 favorites]

I'm with wanderingmind: 1D is boring, 2D is interesting and non-trivial, so I'd start there. Later 2D can be described as a subspace of 3D. The idea is to build people's intuition with something that is a real honest linear space, so I'd stay away from color mixing for a bit. Let people navigate physical space, which is what they've been doing all their lives, then you can give them other things to think about later. Here's a sketch off the top of my head of how I'd play with this given a TA section or 2.

First, orthogonal:

Everyone can navigate Manhattan, given directions, yeah? There are two nice basis vectors (North and East), that I think everyone gets. People can see that there's no way you can describe east in terms of north (without rotation), and vice versa, but with both you can describe every cardinal direction, in a way you couldn't if all you had was a "West arrow" and an "East arrow". There's loads of examples in Real City Life you can use to discuss normalization, superposition, all applied linear algebra fun. So you can play with different representations using orthogonal 2-d basis.

Then, linear independence:

Can you get everywhere if someone gives you NEE and East? Sure, all you need is a little of the north, and you can still get everywhere. Important bit is the spanning of the space.

But what about getting to the 21st floor? We need a new linearly independent vector to get to 3D (2D was a subspace etc). If these were entry-physicists, this would be the time to introduce the Minkowski metric, and Time for 4D, social scientists should know this too, but you might have a rebellion on your hands.

So after people have this language for talking about physical space, let them relax and encourage them to start thinking about everything in terms of spanning basis (personality === Cambell's archetypes. taste === salty, sweet, etc. fashion, music, ..., everything you can describe you can whip up some sort of basis for, although for some things (functions of real variables, eg.) you need infinite bases). Some of these things don't really encourage the analogy of linear superposition, some do, and it's valuable for scientists (even the social kind) to learn the boundaries of the language their using.

Start with the physical though.

posted by johnjoe at 12:11 PM on March 31, 2008 [2 favorites]

I think that there are very few concepts in elementary linear algebra that can't be visualized in two or three dimensions.

*linear independence, subspaces, basis and dimension*

Draw R^2. Two vectors are linearly independent if they point in different directions. A subspace is any line through the origin. A basis for that subspace is any nonzero vector pointing in the direction of that line. Maybe move to R^3 to illustrate the difference between subspaces of dimension 1 and 2.

That is about as concrete as it gets.

posted by number9dream at 6:39 PM on March 31, 2008

Draw R^2. Two vectors are linearly independent if they point in different directions. A subspace is any line through the origin. A basis for that subspace is any nonzero vector pointing in the direction of that line. Maybe move to R^3 to illustrate the difference between subspaces of dimension 1 and 2.

That is about as concrete as it gets.

posted by number9dream at 6:39 PM on March 31, 2008

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posted by wanderingmind at 7:43 PM on March 30, 2008