Hi guys/girls,
just wondering if I could get some help with the following questions I have encountered, any input is appreciated!!
the question considers upper-triangular matrices(All entries below the diagonal are 0), symmetric matrices(A=A transpose), skew-symmetric matrices( A = (-)A transpose) and diagonal matrices(all entries are 0 except those of the diagonal).
The first question is to state whether each of these special matrices (obviously they are all square) are subspaces of M(n*n) - ie square matrices with real numbers. I believe they all are, since each is closed under addition and multiplication, and each can obviously contain 0 (ie a zero matrix)
Secondly it asks if S(2)- ie the respective two by two matrix, is a subspace of the 2*2 matrices,write down a basis for it. (THIS IS THE PART IM NOT SURE OF)
1) For upper triangular I believe it to be easy, since all upper triangular are in echlon form, the column vectors are thus linearly independent, and also span the subspace, hence I just used the columns from the most basic upper triangular with 1's and 0's for entries as my basis
2)For symmetric, I wrote a symmetric matrix down, and reduced to echelon form, and used againt the columns which are linearly independent, as a basis.
I did the same for the remianing 2, using the same technique. (BUT I DONT THISNK ITS CORRECT FOR IT SEEMS TO SIMPLE)
The final quesiton is, given S(n) is a subspace for each of the listed matrices, what is the dimension of each
From my technique above, i will simply get that the Dim = n for each type.
I think this is far to simple and thus I know something im doing isnt correct, could someone please shed some light as to where my miunderstanding is!
Thanks heaps!
just wondering if I could get some help with the following questions I have encountered, any input is appreciated!!
the question considers upper-triangular matrices(All entries below the diagonal are 0), symmetric matrices(A=A transpose), skew-symmetric matrices( A = (-)A transpose) and diagonal matrices(all entries are 0 except those of the diagonal).
The first question is to state whether each of these special matrices (obviously they are all square) are subspaces of M(n*n) - ie square matrices with real numbers. I believe they all are, since each is closed under addition and multiplication, and each can obviously contain 0 (ie a zero matrix)
Secondly it asks if S(2)- ie the respective two by two matrix, is a subspace of the 2*2 matrices,write down a basis for it. (THIS IS THE PART IM NOT SURE OF)
1) For upper triangular I believe it to be easy, since all upper triangular are in echlon form, the column vectors are thus linearly independent, and also span the subspace, hence I just used the columns from the most basic upper triangular with 1's and 0's for entries as my basis
2)For symmetric, I wrote a symmetric matrix down, and reduced to echelon form, and used againt the columns which are linearly independent, as a basis.
I did the same for the remianing 2, using the same technique. (BUT I DONT THISNK ITS CORRECT FOR IT SEEMS TO SIMPLE)
The final quesiton is, given S(n) is a subspace for each of the listed matrices, what is the dimension of each
From my technique above, i will simply get that the Dim = n for each type.
I think this is far to simple and thus I know something im doing isnt correct, could someone please shed some light as to where my miunderstanding is!
Thanks heaps!