Calling for conceptual help!

7 Oct

I know I’m a nerd when a question like the following is my hot topic of the day:

Why is it OK to replace a row in a matrix with the answer from adding two rows?”

For example, in solving for:

x + y + z = 0
x – y + z = 2
x – y – z = 10

a matrix like this can be used:

1   1   1   0
1  -1   1   2
1  -1  -1  10

The next step would be to add two of the rows and replace one of the equations

1   1   1   0
2  0   2   2
1  -1  -1  10

But WHY is this allowed? It’s not like I’m replacing the row with an equivalent equation…

After some discussion and thought, a coworker and I came upon one conceptual theory. Conceptually, what we are doing is replacing an equation of a line with a new equation that got rid of one (or two) of the variables. Each of the variables, in this case, represent an added dimension.

Therefore, what I think we are doing when replacing a row is that we’re getting rid of dimensions and using a model in a lower dimension to represent the original line.

For example, this is how I see it:
Picture a line sticking diagonally out from your computer screen. This line is definitely in 3D. Now imagine taking a picture of that line coming out of the screen. This picture on your camera now made a 3D line into a new 2D line!

It’s kind of like taking a 3D object and taking a picture of that object. That picture is a 2-dimensional representation of the 3D object.

This object was ORIGINALLY in 3-dimensions

This object was ORIGINALLY in 3-dimensions

Taking a picture of it, however, makes this new representation 2D

Taking a picture of it, however, makes this new representation of the same cube, 2D

I think the same thing is happening with these lines. When solving a system of equations like

x + y + z = 0
x – y + z = 2
x – y – z = 10

what we are doing when looking for a “solution” is finding the point of intersection of the 3 lines. So in order to do this, I think we’re taking “pictures” of the lines to simplify our 3D model to 2D and then to a 1D representation that reveals the point of intersection.

So again, this is only what I’ve theorized so far. I can picture it and I’m looking for a way to visually prove it. I’ve never actually taught this before in Algebra 2, always due to lack of time. I finally decided to teach it this year and would love any kind of feedback!

QUESTIONS:

  1. Does this mean I can replace a row only with a new equation that gets rid of a variable/ dimension? What does it mean if I replace a row with a combined equation that does NOT get rid of a variable? Is this like taking a picture from a different angle not parallel to one of the axis?
  2. If all this is true, why does COMBINING the two equations give you an accurate “picture” of the 3D model?
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4 Responses to “Calling for conceptual help!”

  1. Aileen October 8, 2009 at 2:17 am #

    Math = Jinna = Sexy. J’adore math!

  2. Jessica October 8, 2009 at 3:03 am #

    This isn’t directly related to this post, but more a tangent off of a comment I saw you made at Sweeney Math re: the Quadratic Formula Rap. You said you use sign while singing Quadratic Formula in “pop goes the weasel”. Do you teach deaf/hard-of-hearing students? I do…I’m looking to network with other math teachers of the deaf. I’m teaching calculus for the first time this year…pretty exciting!

  3. talkingninja October 9, 2009 at 4:36 pm #

    Hi Jessica,
    I actually don’t teach any deaf or hard of hearing. A colleague of mine taught me the formula and it turned out that most of the kids loved the change and even the challenge of signing for the first time. If you have any other suggestions for a math lesson in sign language I would love to try it! (I have a couple deaf studies friends who can help show me the actual signs)
    Your Calc class sounds like fun already. Looking forward to reading more!

  4. Andrew October 11, 2009 at 9:14 am #

    This is interesting because you’re touching upon the fundamentals of linear algebra but thinking of it from a different perspective. your take on the dimension as it relates to matrices is close to but not quite the rigorous algebraic definition. the reason why you can add/subtract rows or linear combinations of each other is it is a shorthand of substituting equations. doing it by row is called takingthe linear combinations of rows…what you’re doing is reducing the rows to in a form that makes it trivial to find the solution to the system. Dimension technically would be length of the basis vectors that make up the solution to the system. In other words it is the number of values it takes to have enough ‘space’ to fully define the system. so i think what you’re talking about is is not quite reducing the dimension more like simplifying the system. in the case where you are not getting rid of a row it just isn’t smart simplifying and all you’re doing is creating another valid linear combination of the system. It’s sorta like what you described about taking a picture but slightly more subtle. you are changing the picture when u take the linear combinations of the rows. But in doing these row reductions you don’t make a more ‘accurate’ picture, you just get the picture reorganized into its most fundamental constituent parts but with still enough information to fully describe the picture.

    Even more interesting is that you could have solutions to systems that are uniquely defined, undefined, or even over defined where there is more than 1 unique solution. in that case the dimension of the solution to the system is smaller than the dimension of the system itself. so many solutions can ‘reside’ in the dimension of the system. the wikipedia has some great info on linear systems that would give you a better idea of linear systems. In engineering we take advantage of having more equations than unknowns or having more measurements than number of modeled variables. Because the real world is chock full of inaccuracies (while math on paper is exact) you can’t really have an accurate single solution.. so you take a few extra valid solutions and find what the minimum error is between the group of solutions as an estimate of what the best solution may be.

    Its crazy to think that the math that I thought was mostly irrelevant 10 years ago is fundamental for my ability now to understand the dynamics of the world i help engineer devices to work in. I need to study more *gasp!*

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