Hey student2011.
You need to setup a matrix system (Ax=b) corresponding to the hyper-planes and then look at the rank of the solution. That will tell whether it even has a solution and if so whether its a point/line/plane/etc.
Dear mathematicians
I have a question regarding the intersection of three dimensional hyperplanes in R^4. I know that the intersection of two planes in R^2 is a line. So the intersection of a finite number of planes is a line and we can thought of the planes as a pages of an open book.
How can we define the intersection of three-dim hyperplanes in R^4. Is it true that the intersection of finite number of hyperplanes is a common plane between them, if so how can we visualise and sketch this.
Thank you in advance
Hey student2011.
You need to setup a matrix system (Ax=b) corresponding to the hyper-planes and then look at the rank of the solution. That will tell whether it even has a solution and if so whether its a point/line/plane/etc.
Did you mean "two planes in R^3"? There cannot be two distinct planes in R^2.
Even assuming you mean R^3, this, itself, is not generally true. The intersection of two planes in R^2 is (1) a line or (2) the empty set or (3) a plane.
So the intersection of a finite number of planes is a line and we can thought of the planes as a pages of an open book.
How can we define the intersection of three-dim hyperplanes in R^4. Is it true that the intersection of finite number of hyperplanes is a common plane between them, if so how can we visualise and sketch this.
Thank you in advance
Yes, you are right Hallsoflvy. I mean the intersection of planes in R^3 not R^2 there is only one plane in R^2. The intersection is either empty or a plane if they are identical, these two cases are trivial, or a line.
Thanks alot Hallsoflvy