Linear Algebra need help with explanation Thanks If S is the

Linear Algebra, need help with explanation. Thanks!

If S is the subspace of M_3(R) consisting of all lower triangular matrices, then dim S If S is the subspace of m_5(R) consisting of all skew-symmetric matrices, then dim S =

Solution

1a)

lower triangular matrices have all entries above the main diagonal equal to 0

Total entries are:3*3=9

Entries on the diagonal =3

HEnce entries above the diagonal are: (9-3)/2=3

So , 3 entries below the diagonal and 3 on the diagonal so total 6 entries. Hend, dim S=6

b)

IN a skew symmetric matrix all entris on the diagonal are equal to 0

And once all entries below the diagonal are set the entreis above the diagonal are also set

Number of entries below the diagonal are: (5^2-5)/2=10

HEnce, dim S=10