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Linear Algebra 9f: Row Switching in Gaussian Elimination
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- čas přidán 27. 10. 2014
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Michael Jordan, the basketball player 6:23 ...? Very funny, Pavel! No doubt you meant Wilhelm Jordan from 1888. As far as I know Wilhelm never played for the Chicago Bulls ... lol.
8:40 I believe the nullspace is supposed to be [0, 0, 0] instead of just [0] since there are 3 columns in the matrix.
Yes, you're exactly right! Sorry about that.
Thank you very much. I was stuck on it for quite a long and there is no better explanation than yours. Cheers
Thx for explaining so crisply (and humorfully) Gaussian Elimination vs Jordan back substitution! so much clearer in person than reading a deecription.
Thank you. So nice to hear kinds words from a colleague!
I love the Michael Jordan joke.
Great job.
When hould he rows were switched I believe the variables y and z also should be switched?
The order of variables [x y z] as written depends only on the order of the columns. If the columns are switched, then the order needs to be changed.
Why the null space is in R1 and not in R3? Since there are 3 columns in the matrix, it should lie in R3, right?
I believe you are correct, and this was an oversight.
Hi, yeah, I'm also taking the Strang course, and still very confused about this switching.
1) Why is it okay to switch rows? Doesn't that change the transformation the matrix applies to the vector (x,y,z)?
2) It seems like this elimination destroys the original matrix. Why is that okay? Will the reduced matrix still be usable after the elimination, or will it no longer apply the same transformation to the vector (x,y,z) as before?
3) Can you only switch consecutive rows, or can you switch any two rows in a matrix at any time?
4) If switching rows means switching the order of the equations in the "row picture," then can you explain what it means (in an intuitive sense), when you switch the rows, from a COLUMN PICTURE perspective? Many thanks.
You ask a very good question, one that used to puzzle me. The way I think of it is this. Each row can be thought of as an equation. If you have two equations, 3x+4y=7, and 2x+5y=9, does it matter what order they're in, when you set to work to solve them? No. You can write either one first, and put the other below it. The matrix and its column space, change (or at least can change) after a row switch, but the null space remains the same.
The x, y and z are just placeholders (They just tell you that something must take the place of the variable)
Wow. Thank you for asking! And also thank you for the two answers by the gentlemen
Great video's, thank you!
ohhhhhhhh
RREF gives you the solution to x y z
you dont have to go halfway to solve for x y z (REF or Upper Triangular)
you can go the complete path to RREF
If we whant to now y (some ph. entry (let say movement to y direction) is it stil same answear for y if we swiching the rows, and not unknowns?
Of course it is.
Michael?
Yes, the procedure was renamed after the 1998 NBA finals.
The correct comment is that when the rows were switched, y and z should also have been switched.
Definitely not. Switching rows can be thought of as switching the order of equations - and that doesn't change the order of the variables.
One could actually switch columns, which would necessitate switching the corresponding unknowns in order to produce an equivalent linear system.
MathTheBeautiful Thank you.