Subspaces of Vector Spaces

• Dec 11th 2013, 10:42 AM
turbozz
Subspaces of Vector Spaces
Let V be some vector space and S be a subset.

a. V=T(3,3) and S is the set of invertible linear transformations.

I know the V represents all linear transformations from R^3 -> R^3, and all linear transformations in S must be invertible, thus they must all be bijective. But I'm unsure how to go about this?
I know the three conditions that establish a subspace - S contains the 0 vector, S is closed under addition, and S is closed under scalar multiplication. Just not sure how to proceed.
• Dec 11th 2013, 11:20 AM
romsek
Re: Subspaces of Vector Spaces
are you trying to show S is a subspace? Let's get that squared away first.
• Dec 11th 2013, 11:29 AM
turbozz
Re: Subspaces of Vector Spaces
Yes, either proving it is or disproving it.
• Dec 11th 2013, 11:45 AM
romsek
Re: Subspaces of Vector Spaces
well you've outlined the conditions that need to be met for a subspace.

Is 0 in your space? .. DOH. 0 isn't very invertible is it? (it's the zero matrix by the way. your space consists of 3x3 matrices)
• Dec 11th 2013, 11:46 AM
Plato
Re: Subspaces of Vector Spaces
Quote:

Originally Posted by turbozz
Yes, either proving it is or disproving it.

Quote:

Originally Posted by turbozz
Let V be some vector space and S be a subset.

a. V=T(3,3) and S is the set of invertible linear transformations.
I know the V represents all linear transformations from R^3 -> R^3, and all linear transformations in S must be invertible, thus they must all be bijective. But I'm unsure how to go about this?
I know the three conditions that establish a subspace - S contains the 0 vector, S is closed under addition, and S is closed under scalar multiplication. Just not sure how to proceed.

What would such a vector look like?
• Dec 11th 2013, 12:05 PM
turbozz
Re: Subspaces of Vector Spaces
I was unsure of what the 0 vector would be in T(2,2). Originally I thought it would be T(0), but I guess since every linear transformation can be expressed as T(x) = Ax, then the zero 3 by 3 matrix represents the zero matrix for T(2, 2). I assume this is true then for any vector space
T(m, n), where the zero vector is the n by m matrix.
• Dec 11th 2013, 12:13 PM
romsek
Re: Subspaces of Vector Spaces
any dimensions.. what is 0? It's the element such that (A + 0) = A

There's only one matrix in the complex field that satisfies that and that's a matrix of all 0s.
• Dec 11th 2013, 02:06 PM
HallsofIvy
Re: Subspaces of Vector Spaces
Equivalently, T is the set of linear transformations from \$\displaystyle R^3\$ to \$\displaystyle R^3\$ so that "0" is the linear transformation such that 0(x, y, z)= (0, 0, 0). Such a linear transformation does not have an inverse because it is not "one to one".
• Dec 11th 2013, 03:54 PM
turbozz
Re: Subspaces of Vector Spaces
Ok thanks and just to verify one last point, its fair to say that the dim(T(m,n)) =dim(R^n*m).
• Dec 11th 2013, 03:57 PM
romsek
Re: Subspaces of Vector Spaces
Quote:

Originally Posted by turbozz
Ok thanks and just to verify one last point, its fair to say that the dim(T(m,n)) =dim(R^n*m).

If T must be invertible then T cannot be non-square. So m=n

If T is invertible then dim(T(n,n)) = n = dim(R^n)
• Dec 11th 2013, 04:20 PM
turbozz
Re: Subspaces of Vector Spaces
But if we just say vector space T(m, n) contains all linear transformations of the form T:R^m -> R^n (regardless of whether they are invertible or not) then dim(T(m,n)) =mn, right? And since dim(R^n*m) = nm, can't one infer the dimensions are the same?
Note, when I write R^n*m I mean the vector space that contains all n by m matrices.
• Dec 11th 2013, 04:29 PM
romsek
Re: Subspaces of Vector Spaces
Quote:

Originally Posted by turbozz
But if we just say vector space T(m, n) contains all linear transformations of the form T:R^m -> R^n (regardless of whether they are invertible or not) then dim(T(m,n)) =mn, right? And since dim(R^n*m) = nm, can't one infer the dimensions are the same?
Note, when I write R^n*m I mean the vector space that contains all n by m matrices.

at best dim(T(m,n)) = min(m,n)
• Dec 11th 2013, 04:39 PM
HallsofIvy
Re: Subspaces of Vector Spaces
Quote:

Originally Posted by romsek
If T must be invertible then T cannot be non-square. So m=n

If T is invertible then dim(T(n,n)) = n = dim(R^n)

No, that's not true. In the vector space, T(n, n), of linear transformations (NOT necessarily invertible) from \$\displaystyle R^n\$ to \$\displaystyle R^m\$, given a basis \$\displaystyle \{v_k\}\$ for \$\displaystyle R^n\$, we can have take as basis the collection of transformations \$\displaystyle A_{ij}\$ defined by \$\displaystyle A_{ij}(v_i)= v_j\$ if but \$\displaystyle A_{ij}(v_k)= 0\$ for \$\displaystyle k\ne i\$. That has dimension \$\displaystyle n^2\$, not n. More generally, T(n, m), the set of linear transformations (NOT necessarily invertible) from \$\displaystyle R^n\$ to \$\displaystyle R^m\$, given basis \$\displaystyle \{u_i}\$ for \$\displaystyle R^n\$ and basis [itex]\{v_j\}[/itex] for \$\displaystyle R^m\$, has basis the collection of transformations \$\displaystyle A_{ij}\$ deined b \$\displaystyle A_{ij}(u_i)= v_j\$ with \$\displaystyle A_{ij}(u_k)= 0\$ if \$\displaystyle k\ne i\$ and so has dimension nm.
• Dec 11th 2013, 04:41 PM
romsek
Re: Subspaces of Vector Spaces
I'm going to have to study this.

Thank you for the correction.