Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I've been given the following as a homework problem:

Find a basis for the following subspace of $F^5$: $$W = \{(a, b, c, d, e) \in F^5 \mid a - c - d = 0\}$$

At the moment, I've been just guessing at potential solutions. There must be a better method than guess and check.

How do I solve this and similar problems?

share|improve this question
    
Note: Since this is a homework problem, I'm not looking for the answer. Just the strategy to solve this and similar problems. –  Casey Patton Jan 27 '12 at 1:09

2 Answers 2

up vote 11 down vote accepted

Let's look at the following example:

$$W = \{ (a,b,c,d)\in\mathbb{R}^4 \mid a+3b-2c = 0\}.$$

The vector space $W$ consists of all solutions $(x,y,z,w)$ to the equation $$x + 3y - 2z = 0.$$

How do we write all solutions? Well, first of all, $w$ can be anything and it doesn't affect any other variable. Then, if we let $y$ and $z$ be anything we want, then that will force $x$ and give a solution. So we have three degrees of freedom: a free choice of $w$, a free choice of $z$, and a free choice of $y$. Then $x$ will be forced. This suggests dimension $3$.

How does the choice of $w$ affect $x$, $y$, and $z$? In absolutely no way. Since choosing $w$ does not affect $x$, $y$, or $z$, this gives the vector $(0,0,0,1)$: the choice of $w$ (the $1$) does not affect the others.

How does the choice of $z$ affect $x$, $y$, and $w$? It doesn't affect $y$ and $w$. But if $z=1$, then $x$ needs to be $2$: that is, we need to get two $x$s for every $z$. This gives the vector $(2,0,1,0)$.

Finally, now does the choice of $y$ affect $x$, $z$, and $w$? It doesn't affect $z$ and $w$ (they are free), but for every $y$, we need to have $-3$ $x$s. That gives the vector $(-3,1,0,0)$.

So a basis for my $W$ consists of $(-3,1,0,0)$, $(2,0,1,0)$, and $(0,0,0,1)$. You can verify that all of them lie in $W$, and that every vector in $W$ can be written as a linear combination of these three in a unique way.

share|improve this answer
    
Great explanation! Looks like you just popped over the 100k reputation mark as well, congratulations. :) –  Samuel Reid Jan 27 '12 at 5:02
    
Incredibly helpful and did exactly what I asked for: gave me a strategy for solving similar problems. Thanks a bunch! –  Casey Patton Jan 27 '12 at 17:34

let v=real number let U=(A,B,C,D) :B-2C=D=0 W (A,B,C,D:A=,B=2C FIND a basis and the dimension of UnW

share|improve this answer
    
It's difficult to tell what you're suggesting, here. Partly, this is because of your formatting (or lack thereof). See this page for some of the basics of formatting. –  Cameron Buie Nov 27 '12 at 18:46

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.