Finding a subspace (possibly intersection of subspace?)by Throwback Tags: intersection, possibly, subspace 

#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: 20,968

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, 07:20 PM

P: 13

....




#4
Nov511, 07:24 PM

Mentor
P: 20,968

Finding a subspace (possibly intersection of subspace?)This stuff, too. where {As,t in ℝ} ℂ M2x2(ℝ) s, t in what? 



#5
Nov511, 07:32 PM

P: 13

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) 



#6
Nov511, 07:38 PM

Math
Emeritus
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Thanks
PF Gold
P: 38,879

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? 



#7
Nov511, 07:49 PM

P: 13

Closure under addition and closure under multiplication?




#8
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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