I showing work for the following questions

[MastersMath12]

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 independent vectors of a real vector space. Show that u + v and u - v are linearly independent. 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

 

[jbunniii]

Hi MastersMath12,

You need to show us what you have attempted so far.

Also, posts in the homework section need to follow a specific format. See here for more details:

https://www.physicsforums.com/showpost.php?p=4021232&postcount=4

In particular, it's not acceptable simply to post your entire homework assignment and expect people to do it for you. Please limit each thread to one problem, using the correct formatting, and showing what you have attempted so far.

 

[MastersMath12]

Sorry, I am new to the forum. Thanks for passing the guidlines along to me.

I have completed and feel very confident in the about questions except these two below

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 independent vectors of a real vector space. Show that u + v and u - v are linearly independent. 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 independent but I am having trouble prooving if the same conclusion would be true over Z2.

I have completed the above questions, but I am still pretty confused on these two. Any help would be greatly appreciated.

 

[jbunniii]

Sorry, I am new to the forum. Thanks for passing the guidlines along to me.

I have completed and feel very confident in the about questions except these two below

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"

What linear maps did you find?

2) Let u and v be two linear independent vectors of a real vector space. Show that u + v and u - v are linearly independent. 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 independent but I am having trouble prooving if the same conclusion would be true over Z2.

What was your proof for the case of the real vector space?