# Help with a few questions

• Sep 24th 2012, 06:45 AM
MastersMath12
Help with a few questions
1) Which of the following are subspaces of R3

a) T = {(x1, x2, x3) | x1x2x3 = 0}
b) T = {(x1, x2, x3) | x1 - x3 = 0}
c) T = {(x1, x2, x3) | x1 = 0}
d) T = {(x1, x2, x3) | x1 = 1}

2) Let U be a k-vector space, where k is any field. Let V be a subspace of U. Assume that dimk V = dimk U. Show that U = V

3) Let u and v be two linear independant vectors of a real vector space. Show that u + v and u - v are linearly independant. Is the same conclusion true if the vector space was over Za.

4) Let T: U--->V be a linear map. Show that T(0u) = 0v

5) Find an example of vector space V and two subspaces W ⊂ V and Z ⊂ V such that Z ∪ W is not a subspace?

6) Show that T = {a + b√3 | a,b∈T } is a field (a subfield of R). Show that M = {a + b√3 | a,b∈ L} is not a field

Any help would be appreciated
• Sep 24th 2012, 09:50 AM
FernandoRevilla
Re: Help with a few questions
Those are too many questions in only one thread. Besides, you should show some work according to the rules of the forum. A little help, so other members can also answer you.

Quote:

Originally Posted by MastersMath12
1) Which of the following are subspaces of R3 a) T = {(x1, x2, x3) | x1x2x3 = 0}

$\displaystyle (1,1,0)$ and $\displaystyle (0,0,1)$ belong to $\displaystyle T$ but $\displaystyle (1,1,0)+(0,0,1)=(1,1,1)\not\in T$, so $\displaystyle T$ is not a subspace of $\displaystyle \mathbb{R}^3.$
• Sep 24th 2012, 09:58 AM
MastersMath12
Re: Help with a few questions
I made a mistake. Those are actually the questions I had answered. I coppied those by accident. Sorry

Sorry, I am new to the forum.

1) Find two linear maps A,B: R2 ---> R2 such that A°B ≠ B°A

I understand how to find the two linear maps, but I am still lost with respect to "such that A°B ≠ B°A"

2) Let u and v be two linear independant vectors of a real vector space. Show that u + v and u - v are linearly independant. Is the same conclusion true if the vector space was over Z2.

For this one, I am just completely confused. I understand the idea of proving they are linearly independant but I am having trouble prooving if the same conclusion would be true over Z2.

Any help would be greatly appreciated.
• Sep 24th 2012, 08:39 PM
FernandoRevilla
Re: Help with a few questions
Quote:

Originally Posted by MastersMath12
1) Find two linear maps A,B: R2 ---> R2 such that A°B ≠ B°A .I understand how to find the two linear maps, but I am still lost with respect to "such that A°B ≠ B°A"

Choose $\displaystyle A(x_1,x_2)=(x_2,0),\;B(x_1,x_2)=(0,x_1).$ Verify that $\displaystyle A\circ B\neq B\circ A.$

Quote:

2) Let u and v be two linear independant vectors of a real vector space. Show that u + v and u - v are linearly independant. Is the same conclusion true if the vector space was over Z2.
It is not true over $\displaystyle \mathbb{Z}_2=\{0,1\}.$ Counterexample: choose $\displaystyle u=(1,0)$ and $\displaystyle v=(0,1)$, then $\displaystyle \{u,v\}$ are linearly independent, but $\displaystyle u+v=(1,1)$ and $\displaystyle u-v=(1,-1)=(1,1)$, so $\displaystyle u + v$ and $\displaystyle u-v$ are not linearly independent.