Vector space dimension question

[P-Jay1]

Prove that the space of 2×2 real matrices forms a vector space of dimension 4 over
R. [12]

Im unsure, anyone any idea?

 

[boaz]

If I understood right, we have to find an isomorphism from M^{R}_{2x2} to R^4.

In other word, we need to find a linear transformation, T:M^{R}_{2x2} \rightarrow R^4 that ker(T)=0.

HINT: look at T[a,b;c,d]=(a,b,c,d)

 

[Fredrik]

What's your book's definition of "dimension 4". I use the definition that says that a vector space is 4-dimensional if it contains a linearly independent set with 4 members, but no linearly independent set with 5 members. So to solve your problem, I would find a set of 4 linearly independent matrices and prove that every matrix is a linear combination of those four.

 

[P-Jay1]

So what does ker(T) stand for? I am still really clueless. So do I find an isomorphism from T[a,b;c,d] to (a,b,c,d)?

 

[Fredrik]

So what does ker(T) stand for? I am still really clueless. So do I find an isomorphism from T[a,b;c,d] to (a,b,c,d)?

ker T is the set of all x such that Tx=0. Finding that isomorphism is one way to do it, but not the only one. Edit: Wait, you said "from T[a,b;c,d]". I don't know what you mean by that. I meant that one way to solve the problem is to find an isomorphism from the set of 2×2 matrices to ℝ⁴.

 

[boaz]

ker T is the set of all x such that Tx=0. Finding that isomorphism is one way to do it, but not the only one. Edit: Wait, you said "from T[a,b;c,d]". I don't know what you mean by that. I meant that one way to solve the problem is to find an isomorphism from the set of 2×2 matrices to ℝ⁴.

I guess I wasn't much clear. What I meant is that T takes a matrix [a,b;c,d] and gives a vector (a,b,c,d). Actually, I think that this is what he has meant.

 

[P-Jay1]

A = (a b c d)

A = a (1 0 0 0) + b (0 1 0 0) + c (0 0 1 0) + d (0 0 0 1)

Please excuse my lack of ability to lay this out correctly but each of these in brackets have 2 rows and 2 columns i.e 2 x 2. not 1 x4

a,b... Eℝ is a vector space over ℝ

Is this the correct approach?

 

[boaz]

it seems like you know the answer but don't know how to do this.
so first of all, let's state that "in order to prove that the space of 2×2 real matrices forms a vector space of dimension 4 over R. we will define an isomorphism from 2x2 real matrices space to R4". Later, we will say. "lets take a look at the following transformation T:M2x2->R4, T([a,b;c,d])=(a,b,c,d)"
now, what is left to do is to show that it is linear transformation and that it is a isomorphism.

regarding the linear transformation, it's not a big deal. i am sure you can handle it by yourself.
to show that T is isomorphism we need to prove that T is a linear transformation and ker(T)=0 or im(T)=4.

to do that - we need to know first what im(T) is equal. if we take a base of one space and we transform it using T we will get the base of the other space we are transforming to. so let's take the standard base of M_2x2(R). { [1,0;0,0], [0,1;0,0], [0,0;1,0], [0,0;0,1]}. from what we have defined before:
T([1,0;0,0])=(1,0,0,0); T([0,1;0,0])=(0,1,0,0) ... T([0,0;0,1])=(0,0,0,1).
in other terms, imT(M_2x2(R) ) = sp{(1,0,0,0),...,(0,0,0,1)}, but the set {(1,0,0,0)...(0,0,0,1)} is the standard base of R4 thus dim(imT)=4. so in conclusion, T is an isomorphism, and we proved what we have asked for.

 

[HallsofIvy]

Any 2 by 2 matrix is of the form
\begin{bmatrix}a & b \\ c & d \end{bmatrix}= a\begin{bmatrix}1 & 0 \\ 0 & 0 \end{bmatrix}+ b\begin{bmatrix}0 & 1 \\ 0 & 0 \end{bmatrix}+ c\begin{bmatrix}0 & 0 \\ 1 & 0 \end{bmatrix}+ d\begin{bmatrix}0 & 0 \\ 0 & 1 \end{bmatrix}
The most "basic" definition of "dimension" is that it is the number of vectors in any space. It is easy to see that these four matrices are a basis.

Or, if as Boaz does, you wish to find an isomorphism,
f\left(\begin{bmatrix}a & b \\ c & d\end{bmatrix}\right)= (a, b, c, d)
works nicely.