Find the basis of this vector space

[preferred_anonymous]

Hi! I am given a Vector space V = { (x,y,z) E R³ | 13x+17y+3z=0 and 13+17y+4z = 0 } and I am asked to find its basis. However, I kinda get stuck with the free variables. If only the first equation had to be true (13x+17y+3z=0), then I could easily say that y,z are free variables, and then compute x,for y = 1 and z = 1 to find the vector space's basis. Now that I have 2 equations however, if I say that z is a free variable, and for z = 1, x = -3/13 and x = -4/13, which of course cannot be true. What can I do to solve this?

 

[Dr.Peterson]

First, I think you need to quote the problem exactly as given to you. "Find its basis" isn't a valid request, as there are typically many sets that can be used as a basis. And that is probably part of your problem, as there are multiple choices you can make.

Also, check for a typo in your equations.

But also, it may be helpful if you try solving the system of equations 13x+17y+3z=0 and 13+17y+4z = 0, in any way you choose. Tell us what you find.

 

[HallsofIvy]

As Dr. Peterson said, a vector space has infinitely many bases. You are asked to find one of those infinitely many bases so you can expect to have arbitrary choices. You are given that 13x+17y+3z=0 and 13x+17y+4z = 0. (You have just "13", not "13x" but that can't be right.)

First, \(\displaystyle R^3\) is, of course, 3 dimensional. Requiring that the coordinates satisfy a given linear equation reduces the dimension by 1. Requiring that they satisfy two equations reduces by 2 so we are looking for a subspace of dimension 3- 2= 1, geometrically, a straight line through the origin.

Subtracting the first equation from the second eliminates both x and y leaving just z= 0. Aha! That tells us the the subspace of R^3 that satisifes both of these equations lies in the xy-plane! With z= 0, both equations become 13x+ 17y= 0. That's a straight line in the xy-plane so is one dimensional so has a basis consisting of a single vector. One easy solution to 13x+ 17y= 0 is x= 17, y= -13. A basis for this subspace is {<17, -13, 0>}.

 

[Steven G]

If you some expression that equals 0 and the same expression + z that equals 0, then z=0. Do you not see that? For the record, the two equations did not have to equal 0 to get z=0, but they needed to equal the same quantity, like 27