Adding Matrix and Polynomial Basis Activity to Address #354 #356
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@StevenClontz I'm tagging you since you're also teaching this upcoming. If we get this merged (possibly after some helpful suggestions), I'd be happy to report how this lesson goes (which I expect will be the week after next). |
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Co-authored-by: Steven Clontz <steven.clontz@gmail.com>
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jkostiuk
changed the title
| <task> | ||
| <statement> | ||
| <p> How many solutions does this equation have? | ||
| <ol marker="A." cols="2"> | ||
| <li>1</li> | ||
| <li>infinitely many</li> | ||
| <li>none</li> | ||
| <li>2</li> | ||
| </ol> | ||
| </p> | ||
| </statement> | ||
| </task> | ||
| <task> | ||
| <statement> | ||
| <p> | ||
| How many solutions do the following matrix equation have? | ||
| <me> | ||
| x_1\left[\begin{array}{cc} | ||
| 1&0\\0&0 | ||
| \end{array}\right]+x_2\left[\begin{array}{cc} | ||
| 0&1\\0&0 | ||
| \end{array}\right]+x_3\left[\begin{array}{cc} | ||
| 0&0\\1&0 | ||
| \end{array}\right]+x_4\left[\begin{array}{cc} | ||
| 0&0\\0&1 | ||
| \end{array}\right]=\left[\begin{array}{cc} | ||
| 4&3\\1&2 | ||
| \end{array}\right]. | ||
| </me> | ||
| <ol marker="A." cols="2"> | ||
| <li>1</li> | ||
| <li>infinitely many</li> | ||
| <li>none</li> | ||
| <li>2</li> | ||
| </ol> | ||
| </p> | ||
| </statement> | ||
| </task> |
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Minor point: the equation in the introduction probably belongs in the first task.
Major point: how do you expect students to answer these questions? They cannot convert the equation to a Euclidean vector equation (that's the point of this activity, to convince them that they can convert to
But what about this? Just ask if (the basis) is a spanning set, with answer choices:
- No, [example] is not a linear combination of these matrices. (3x)
- Yes, every matrix in M22 is a linear combination of these matrices.
Next part: is (the basis) linearly independent?
- No, [matrix] is a linear combination of [matrices] (3x)
- Yes, each matrix cannot be expressed as a linear combination of other matrices in the set.
Then you're set up for part (c) without having to solve equations.
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So, what I coach them to do in the past is to simplify the LHS of that matrix equation into matrix(x1,x2;x3,x4) and then we're just looking at matrix(x1,x2;x3,x4)= some specific matrix.
My students seem to have a hard time really processing how to build a linear combination of matrices and so I think having the LHS written out gives them a starting point.
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ultimately, I think they'll be forced do what I describe (or take a more flexible approach) with your scaffolding, so I'm tempted to give it a shot. I also think your scaffolding would lead to more diversity of discussion in a productive way.
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Oh of course. I too overlooked the obvious approach to turn [matrix]=[matrix] into a linear system rather than trying to find the right Euclidean vector equation. So scaffolding in that direction works as well, but I think the made up constants on the right-hand side is distracting more than building on their intuition of "spanning means we can build everything" and "independence means there's no redundant information" to approach this.
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agreed. i've got a commit coming in with the suggested changes for your review
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I like the idea of having students develop the bases for |
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Here is my suggestion for #354
AT6 relies upon us having bases for our vector spaces. I decided to structure the discussion around our vector-equations to help reinforce how "all we're doing" is using our old stuff in a new context. We can do this if we have a basis and so, first, we need to justify we have a basis.