Showing a set of matrices is a direct sum.

[trap101]

Let W₁ = {A$\in$ M_nXn(R)| A = A^T} and W₂ = {A$\in$ M_nXn(R)| A = -A^T}

Show that M_nXn = W₁ (+) W₂

where the definition of direct sum is:

V is the direct sum of W₁ and W₂ in symbols:

V = W₁ (+) W₂ if:

V = W₁ + W₂ and
W₁ $\cap$ W₂ = {0}


Attempt:

I figure I have to show each property individually. So for the first property I tried to do a manipulation:

A^T + (-A^T) = (A + (-A))^T = 0^T

Then I added A to each side: A + (A + (-A))^T = A + 0^T

Did I even show what was needed?

 

[HallsofIvy]

No, you appear to be misunderstanding what "W₁= {A$\in$ M_xn|A= A^T}" and "W₂= {A$\in$ M_nxn|A= -A^{T[/sub]}" mean. Members of V are of the form A+ B where A is in W₁ and B is in W₂. You cannot use the same matrix, A, for each.}

 

[trap101]

No, you appear to be misunderstanding what "W₁= {A$\in$ M_xn|A= A^T}" and "W₂= {A$\in$ M_nxn|A= -A^{T[/sub]}" mean. Members of V are of the form A+ B where A is in W₁ and B is in W₂. You cannot use the same matrix, A, for each.}

<sup>How's this:

Letting A be a matrix from W1 and B be a matrix from W2:

V = A + B
= AT+ (-BT)
= (A+(-B))T

Can I bring out the -1 and have: (-1)(A+B)T?</sup>