Well i will try to help... ( my english is bad so if you dont understand some words just tell and i will try to explain better )

In a) part you need to give a basis for V.

So the "best" basis for any space is canon basis

( example.

for space R^2 the best basis is:

{ ( 1, 0 ) , ( 0 , 1 ) }

)

So in this case the best basis would be:

1 0 0

0 0 0 = A1

0 0 0

0 1 0

0 0 0 = A2

0 0 0

0 0 1

0 0 0 = A3

0 0 0

0 0 0

0 1 0 = A4

0 0 0

0 0 0

0 0 1 = A5

0 0 0

0 0 0

0 0 0 = A6

0 0 1

to show that this set ( lets name it S ) of matrices is really basis for space of upper triangular matrices you need to show this:

if you have any upper triangular matrix:

x1 x2 x3

0 x4 x5 = X

0 0 x6

you can denote it like:

X = x1*A1 + x2*A2 + x3*A3 + x4*A4 + x5*A5 + x6*A6

so this is the proof that you can get any upper triangular matrix with linear combination of matrix from set S.

now you need to show that: ( y1, ... y6 ----> some variables )

0 0 0

0 0 0 = y1*A1 + y2*A2 + ... + y6*A6

0 0 0

is only on trivial way...

that is easy ( you need to show that this is true only if y1 = y2 = ... y6=0)

and that is the proof that S is basis for space of upper triangular matrices

c) iv

dont really know what you need to do ( dont understand ), but it cant be hard and if that is what i think it is than you will know what to do ( if you really know to solve problems c)i, ii, iii )