Show that the sets of upper-triangular and lower-triangular matrices are subspaces of f n, n . call them u and l respectively. what are their dimensions?

dim L = dim U =

Step-by-step explanation:

We can do it only for the lower-triangular matrices, the case of the upper-triangular matrices is similar. We might caracterice nxn the lower-triangular matrices, as the nxn matrices such that the entry if i<j.

Now let and be two lower triangular matrices, now if

for some

then the entry of C is equal to

Now, if i<j, it must hold that . Therefore, if this is the case we must have that and so we get that C is also a lower triangular matrix. This showa that L is closed under sum and scalar multiplcation, hence it is a linear subspace.

To find the dimension, note that all the entries of a lower-triangular matrix over the diagonal must be equal to zero. However, each entry of the matrix under the diagonal and in the diagonal might be any element of , any entry that can be choosen add up to the dimension of L, we n such elemnts for the first column, (n-1) for the second column, (n-2) for the third column etc.... Therefore,