Linear algebra. Problem with vector spaces dimension

[sphlanx]

Homework Statement

First of all sorry if my terminology sounds a bit weird, i have never studied mathematics in english before.

So this is the problem: We have the space R^2x2 of all the tables with numbers in R. We also have a subspace V of R^2x2 of all the tables with the following property: "If x,y is the first row and z,w the second row then: 3x+8y+5z+w=0. The main question is what is the dimV? There are more questions but i think i can solve them if I know dimV!
//EDIT: I will add a few more questions that i see i have a hard time solving even if I take into account that dimV=3
a) How to prove that dimV=3(i believe it is 3 because the subspace of the solution set of the linear system I provided is 3dimensonial)
b) Is there any other subspace of R^2x2 different BUT isomorphic with V
c) If D a subspace of R^2x2 with dimD=4 then D "contains" V
d) Is there any other subspace of R^2x2, isoporphic to V, that intersected with V has only one element, the zero element.

Homework Equations

The Attempt at a Solution

a)I can see that with the given equation not all of the variables are linear independent. This makes me think that dimV=3 but i can't figure out a way to prove it!
b)I believe the answer is YES. In a previous homework there was the same question about 2-dimensional subspaces and I replied yes because I can think of 2 planes crossing. I am not sure if 3 dimensional spaces can "cross" though.
c) Not a clue :P

Thanks in advance! (the deadline is tommorow Saturday at 24:00 so i would appreciate a quick answer!)

 

[HallsofIvy]

Homework Statement

First of all sorry if my terminology sounds a bit weird, i have never studied mathematics in english before.

So this is the problem: We have the space R^2x2 of all the tables with numbers in R. We also have a subspace V of R^2x2 of all the tables with the following property: "If x,y is the first row and z,w the second row then: 3x+8y+5z+w=0."

The correct word is "matrices", not "tables". "Arrays" and "tables" do not have the addition and multiplication operations that matrices do and addition and scalar multiplication are necessary for a vector space.

The main question is what is the dimV? There are more questions but i think i can solve them if I know dimV!
//EDIT: I will add a few more questions that i see i have a hard time solving even if I take into account that dimV=3
a) How to prove that dimV=3(i believe it is 3 because the subspace of the solution set of the linear system I provided is 3dimensonial)

The given vector space of 2 by 2 matrices is, of course, 4 dimensional and, generally, one equation restricting the components reduces the dimension by 1. More precisely, you can solve the given equation for one of the components, say w= -3x- 8y- 5z. Then you can write the matrices as
$$\begin{bmatrix}x & y \\ z & w\end{bmatrix}= \begin{bmatrix}x & y \\ z & -3x- 8y- 5z\end{bmatrix}$$
$$= \begin{bmatrix}x & 0 \\ 0 & -3x\end{bmatrix}+ \begin{bmatrix}0 & y \\ 0 & -8y\end{bmatrix}+ \begin{bmatrix}0 & 0 \\ z & -5z\end{bmatrix}$$
$$= x\begin{bmatrix}1 & 0 \\ 0 & -3\end{bmatrix}+ y\begin{bmatrix}0 & 1 \\ 0 & -8\end{bmatrix}+ z\begin{bmatrix}0 & 0 \\ 1 & -5\end{bmatrix}$$

b) Is there any other subspace of R^2x2 different BUT isomorphic with V

All subspaces of the same dimension are isomorphic. Try just moving the components around.

c) If D a subspace of R^2x2 with dimD=4 then D "contains" V

Unless I have completely misunderstood what "R^2x2", the only subspace of dimension 4 is R^2x2 itself.

d) Is there any other subspace of R^2x2, isoporphic to V, that intersected with V has only one element, the zero element.

If V has dimension 3 than any subspace that intersects V only at the zero element must have dimension 1.

Homework Equations

The Attempt at a Solution

a)I can see that with the given equation not all of the variables are linear independent. This makes me think that dimV=3 but i can't figure out a way to prove it!
b)I believe the answer is YES. In a previous homework there was the same question about 2-dimensional subspaces and I replied yes because I can think of 2 planes crossing. I am not sure if 3 dimensional spaces can "cross" though.
c) Not a clue :P

Thanks in advance! (the deadline is tommorow Saturday at 24:00 so i would appreciate a quick answer!)