Second Day
Aug. 20th, 2019 07:20 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowLinear algebra is, for me, an interesting course to teach. One of my fundamental maxims of mathematics is this: "Algebra yields information. Geometry yields understanding." (I sometimes call it "Stoutfellow's First Law".) Linear algebra is a hybrid field, which can be viewed algebraically or geometrically, and I always push my students to be prepared to view it either way.
Today, we discussed systems of linear equations and the method of Gaussian elimination. GE is a procedure for changing a system of equations without changing the solutions, pushing it towards a form where the solutions are easy to compute. To illustrate how it works, I wrote down a pair of equations, say, x + y = 3, -x + 4y = 5. (Those aren't the equations I used, but they'll do.) Then I drew coordinate axes and the lines corresponding to those equations: two lines, intersecting in a single point, and that point's coordinates are the unique solution to the equations. Then I began working the transitions.
"The first operation of GE is to swap the equations - put them in a different order. What does that do to the geometry?"
"Nothing."
"Right - the lines get relabeled, but nothing else changes."
"The second operation is to multiply one of the equations, say equation two, by a nonzero constant. What happens to the geometry?"
(Some fumbling, a student talking to himself): "Nothing."
"Right; if you multiply an equation by a nonzero constant, its solutions don't change."
"The third operation is to add a multiple of one equation to another." (I do an example, and draw the new line.) "Line two rotates around the intersection point. If I add different multiples of equation one, line two rotates to different places. In particular, making the right choice, I can rotate it to be horizontal. What does that mean algebraically?"
(Hem, haw.) "The variable x no longer appears in that equation."
"Yes!" (I then point out the connection between this and the algebraic concept of "row echelon form". With many gestures, I try to display what happens if you're working with equations in three variables instead of two.)
Algebra yields information. Geometry yields understanding. I hope I can drive that through to my students.
Today, we discussed systems of linear equations and the method of Gaussian elimination. GE is a procedure for changing a system of equations without changing the solutions, pushing it towards a form where the solutions are easy to compute. To illustrate how it works, I wrote down a pair of equations, say, x + y = 3, -x + 4y = 5. (Those aren't the equations I used, but they'll do.) Then I drew coordinate axes and the lines corresponding to those equations: two lines, intersecting in a single point, and that point's coordinates are the unique solution to the equations. Then I began working the transitions.
"The first operation of GE is to swap the equations - put them in a different order. What does that do to the geometry?"
"Nothing."
"Right - the lines get relabeled, but nothing else changes."
"The second operation is to multiply one of the equations, say equation two, by a nonzero constant. What happens to the geometry?"
(Some fumbling, a student talking to himself): "Nothing."
"Right; if you multiply an equation by a nonzero constant, its solutions don't change."
"The third operation is to add a multiple of one equation to another." (I do an example, and draw the new line.) "Line two rotates around the intersection point. If I add different multiples of equation one, line two rotates to different places. In particular, making the right choice, I can rotate it to be horizontal. What does that mean algebraically?"
(Hem, haw.) "The variable x no longer appears in that equation."
"Yes!" (I then point out the connection between this and the algebraic concept of "row echelon form". With many gestures, I try to display what happens if you're working with equations in three variables instead of two.)
Algebra yields information. Geometry yields understanding. I hope I can drive that through to my students.
