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,

7. d. 64

8. i. 32

9. c. 125

10. j. none of the above

11. d. 81

12. i. 216

13. e. none of the above

14. g. 240

15. e. none of the above

16. g. 32

17.   e. none of the above

where's the diagram?

step-by-step explanation: