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Finding a subspace (possibly intersection of subspace?) 
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#1
Nov511, 05:31 PM

P: 13

1. The problem statement, all variables and given/known data
Let A be the following 2x2 matrix: s 2s 0 t Find a subspace B of M_{2x2} where M_{2x2} = A (+) B 2. Relevant equations A ∩ B = {0} if u and v are in M_{2x2}, then u + v is in M_{2x2} if u is in M_{2x2}, then cu is in M_{2x2} 3. The attempt at a solution Let B be the following 2x2 matrix: 0 0 r 0 Because they are both subspace, they intersect at the zero vector and thus the set {0}, the zero subspace, is a subspace of M_{2x2}. We then have M_{2x2} = A (+) B: M_{2x2} = A + B /\ A ∩ B = {0} 


#2
Nov511, 07:17 PM

Mentor
P: 21,215

It also doesn't make sense to add a matrix  A  and a subspace  B. What is the exact wording of this problem? 


#3
Nov511, 07:24 PM

Mentor
P: 21,215

This stuff, too. where {As,t in ℝ} ℂ M2x2(ℝ) s, t in what? 


#4
Nov511, 07:32 PM

P: 13

Finding a subspace (possibly intersection of subspace?)
This should make it easier haha
In case the image isn't showing either, the symbol that isn't showing is the "R" for real numbers, so s,t in R and M(R) 


#5
Nov511, 07:38 PM

Math
Emeritus
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Thanks
PF Gold
P: 39,348

I take it then that you mean B is a subspace of the space of all two by two matrices with real entries. However, you do NOT mean that A is the "matrix" given. Rather, A is the subspace of all two by two matrices, with real entries, of the form
[tex]\begin{bmatrix}s & 2s \\ 0 & t\end{bmatrix}[/tex]. Saying that "[itex]A(+)B= M_{22}(R)[/itex]" means that for any numbers u, x, y, z, there exist numbers a, b, c, d such that [tex]\begin{bmatrix}s & 2s \\ 0 & t\end{bmatrix}+ \begin{bmatrix}a & b \\ c & d\end{bmatrix}= \begin{bmatrix}u & x \\ y & z\end{bmatrix}[/tex] Of course, then we must have [tex]\begin{bmatrix} a & b \\ c & d\end{bmatrix}= \begin{bmatrix}u s & x 2s \\ y & z t\end{bmatrix}[/tex] Now, what relations must a, b, c, and d satisfy? 


#6
Nov511, 07:49 PM

P: 13

Closure under addition and closure under multiplication?



#7
Nov511, 08:00 PM

P: 13

I barely see what it's asking...
Given A, I don't have a problem proving that A is a subspace of M_{22}  just show there's closure under addition and multiplication. I can find a basis/span, etc. For this question, I'm somewhat lost. Since A + B = M_{22}, then B = M_{22}  A, I'm assuming B has to follow the closure requirements, but is that it? It seems like I'm just missing something pretty big here... 


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