Triangular matricies and subspaces

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mohdhm
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hello again

I was asked if the set of all uppertriangular nxn matricies are a subspace of Mnn,

how would you check if it has a zero vector and closed under addition and multiplication ? and why did they ask for the upper triangular matrix instead of the lower one? or either
 

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Ask yourself the following:

1) If all the upper-triangular elements were to take on values of zero, is the zero matrix contained in the subspace (and is this the same zero vector of your vector space? This means it should be an n by n zero matrix).

2) If you add an upper-triangular matrix with another upper-triangular matrix, do you get an upper-triangular matrix in return? Does the elements of the matrix have elements from your field?

3) Do this again with multiplying your upper-triangle matrix by a constant. Do you get another upper-triangular matrix? Does the elements of the matrix exist in your field?

Do this for lower-triangular ones and diagonal ones and see if they're subspaces or not.
 
thanks brian, makes sense, and so simple