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
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.
Choose Verify that
It is not true over Counterexample: choose and , then are linearly independent, but and , so and are not linearly independent.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.