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On one of my exams last year, we were given a problem (we chose five or six out of eight problems) on an exam, the goal of which was to prove the Bruhat decomposition for $GL_n(k)$. I was one of the two people to choose said problem. I gave a very long convoluted argument which although correct was really inelegant. I proved it more than once because I wasn't satisfied with my proof, and I figured out a somewhat slick contradiction argument based on maximizing leading zeroes of rows (number of zeroes before the pivot), but the proof was still a real mess.

Statement of the problem:

Let $G:=GL(V)$ for $V$ a finite dimensional $k$ vector space. Let $B$ be the stabilizer of the standard flag (these will be invertible upper triangular matrices), and let $W$ be the subgroup of permutation matrices.

Show that $G=\coprod_{w\in W} BwB$, where the $BwB$ are double cosets. That is, show that $G=BWB$.

Question: Is there a slick proof of this fact maybe using "more machinery"? In particular, is there any sort of "coordinate-free" proof (Can we even define the Borel and Weyl subgroups without coordinates?)?

Edit: I take issue with Kevin Buzzard's comment mainly for the last line. It makes me look like a jerk and a hypocrite, but that's not a fair characterization. First of all, I've already proven this. I was asking for a better proof. That's more "due diligence" than any question I've voted down for being in a standard reference can claim. Additionally, how can you honestly expect someone who's never seen the term "reductive group" before to be familiar with the references? I searched for something like proof of Bruhat Decomposition of GL_n on Google, Google books, and Wikipedia and was not able to find a reference. Granted, the last time I searched for it was right after the exam (probably a little less than a year ago), but that still doesn't mean that I failed to fulfill the necessary prerequisites for asking a question on MO.

I feel like people have a knee-jerk response whenever they see me on MO, and I don't think many people here are willing to give me the benefit of the doubt. However, I think I hold myself to the same standards I hold other people to, and I find it in bad taste for someone to challenge my integrity like that. I'd rather delete questions that have accusations like that than have to deal with them like this, because it's embarrassing that I have to keep doing things like this. I don't want to stand out as a whiner, but there are certain accusations that I have to respond to.

On one of my exams last year, we were given a problem (we chose five or six out of eight problems) on an exam, the goal of which was to prove the Bruhat decomposition for $GL_n(k)$. I was one of the two people to choose said problem. I gave a very long convoluted argument which although correct was really inelegant. I proved it more than once because I wasn't satisfied with my proof, and I figured out a somewhat slick contradiction argument based on maximizing leading zeroes of rows (number of zeroes before the pivot), but the proof was still a real mess.

Statement of the problem:

Let $G:=GL(V)$ for $V$ a finite dimensional $k$ vector space. Let $B$ be the stabilizer of the standard flag (these will be invertible upper triangular matrices), and let $W$ be the subgroup of permutation matrices.

Show that $G=\coprod_{w\in W} BwB$, where the $BwB$ are double cosets. That is, show that $G=BWB$.

Question: Is there a slick proof of this fact maybe using "more machinery"? In particular, is there any sort of "coordinate-free" proof (Can we even define the Borel and Weyl subgroups without coordinates?)?

On one of my exams last year, we were given a problem (we chose five or six out of eight problems) on an exam, the goal of which was to prove the Bruhat decomposition for $GL_n(k)$. I was one of the two people to choose said problem. I gave a very long convoluted argument which although correct was really inelegant. I proved it more than once because I wasn't satisfied with my proof, and I figured out a somewhat slick contradiction argument based on maximizing leading zeroes of rows (number of zeroes before the pivot), but the proof was still a real mess.

Statement of the problem:

Let $G:=GL(V)$ for $V$ a finite dimensional $k$ vector space. Let $B$ be the stabilizer of the standard flag (these will be invertible upper triangular matrices), and let $W$ be the subgroup of permutation matrices.

Show that $G=\coprod_{w\in W} BwB$, where the $BwB$ are double cosets. That is, show that $G=BWB$.

Question: Is there a slick proof of this fact maybe using "more machinery"? In particular, is there any sort of "coordinate-free" proof (Can we even define the Borel and Weyl subgroups without coordinates?)?

Edit: I take issue with Kevin Buzzard's comment mainly for the last line. It makes me look like a jerk and a hypocrite, but that's not a fair characterization. First of all, I've already proven this. I was asking for a better proof. That's more "due diligence" than any question I've voted down for being in a standard reference can claim. Additionally, how can you honestly expect someone who's never seen the term "reductive group" before to be familiar with the references? I searched for something like proof of Bruhat Decomposition of GL_n on Google, Google books, and Wikipedia and was not able to find a reference. Granted, the last time I searched for it was right after the exam (probably a little less than a year ago), but that still doesn't mean that I failed to fulfill the necessary prerequisites for asking a question on MO.

On one of my exams last year, we were given a problem (we chose five or six out of eight problems) on an exam, the goal of which was to prove the Bruhat decomposition for $GL_n(k)$. I was one of the two people to choose said problem. I gave a very long convoluted argument which although correct was really inelegant. I proved it more than once because I wasn't satisfied with my proof, and I figured out a somewhat slick contradiction argument based on maximizing leading zeroes of rows (number of zeroes before the pivot), but the proof was still a real mess.

Statement of the problem:

Let $G:=GL(V)$ for $V$ a finite dimensional $k$ vector space. Let $B$ be the stabilizer of the standard flag (these will be invertible upper triangular matrices), and let $W$ be the subgroup of permutation matrices.

Show that $G=\coprod_{w\in W} BwB$, where the $BwB$ are double cosets. That is, show that $G=BWB$.

Question: Is there a slick proof of this fact maybe using "more machinery"? In particular, is there any sort of "coordinate-free" proof (Can we even define the Borel and Weyl subgroups without coordinates?)?

Edit: I take issue with Kevin Buzzard's comment mainly for the last line. It makes me look like a jerk and a hypocrite, but that's not a fair characterization. First of all, I've already proven this. I was asking for a better proof. That's more "due diligence" than any question I've voted down for being in a standard reference can claim. Additionally, how can you honestly expect someone who's never seen the term "reductive group" before to be familiar with the references? I searched for something like proof of Bruhat Decomposition of GL_n on Google, Google books, and Wikipedia and was not able to find a reference. Granted, the last time I searched for it was right after the exam (probably a little less than a year ago), but that still doesn't mean that I failed to fulfill the necessary prerequisites for asking a question on MO.

I feel like people have a knee-jerk response whenever they see me on MO, and I don't think many people here are willing to give me the benefit of the doubt. However, I think I hold myself to the same standards I hold other people to, and I find it in bad taste for someone to challenge my integrity like that. I'd rather delete questions that have accusations like that than have to deal with them like this, because it's embarrassing that I have to keep doing things like this. I don't want to stand out as a whiner, but there are certain accusations that I have to respond to.

On one of my exams last year, we were given a problem (we chose five or six out of eight problems) on an exam, the goal of which was to prove the Bruhat decomposition for $GL_n(k)$. I was one of the two people to choose said problem. I gave a very long convoluted argument which although correct was really inelegant. I proved it more than once because I wasn't satisfied with my proof, and I figured out a somewhat slick contradiction argument based on maximizing leading zeroes of rows (number of zeroes before the pivot), but the proof was still a real mess.

Statement of the problem:

Let $G:=GL(V)$ for $V$ a finite dimensional $k$ vector space. Let $B$ be the stabilizer of the standard flag (these will be invertible upper triangular matrices), and let $W$ be the subgroup of permutation matrices.

Show that $G=\coprod_{w\in W} BwB$, where the $BwB$ are double cosets. That is, show that $G=BWB$.

Question: Is there a slick proof of this fact maybe using "more machinery"? In particular, is there any sort of "coordinate-free" proof (Can we even define the Borel and Weyl subgroups without coordinates?)?

On one of my exams last year, we were given a problem (we chose five or six out of eight problems) on an exam, the goal of which was to prove the Bruhat decomposition for $GL_n(k)$. I was one of the two people to choose said problem. I gave a very long convoluted argument which although correct was really inelegant. I proved it more than once because I wasn't satisfied with my proof, and I figured out a somewhat slick contradiction argument based on maximizing leading zeroes of rows (number of zeroes before the pivot), but the proof was still a real mess.

Statement of the problem:

Let $G:=GL(V)$ for $V$ a finite dimensional $k$ vector space. Let $B$ be the stabilizer of the standard flag (these will be invertible upper triangular matrices), and let $W$ be the subgroup of permutation matrices.

Show that $G=\coprod_{w\in W} BwB$, where the $BwB$ are double cosets. That is, show that $G=BWB$.

Question: Is there a slick proof of this fact maybe using "more machinery"? In particular, is there any sort of "coordinate-free" proof (Can we even define the Borel and Weyl subgroups without coordinates?)?

Edit: I take issue with Kevin Buzzard's comment mainly for the last line. It makes me look like a jerk and a hypocrite, but that's not a fair characterization. First of all, I've already proven this. I was asking for a better proof. That's more "due diligence" than any question I've voted down for being in a standard reference can claim. Additionally, how can you honestly expect someone who's never seen the term "reductive group" before to be familiar with the references? I searched for something like proof of Bruhat Decomposition of GL_n on Google, Google books, and Wikipedia and was not able to find a reference. Granted, the last time I searched for it was right after the exam (probably a little less than a year ago), but that still doesn't mean that I failed to fulfill the necessary prerequisites for asking a question on MO.