Uni level linear algebra question.

[winteroh]

Hi, i need help with my algebra question,
I have done all the questions but im not sure whether its right or wrong please help me
Question:
Let W=span (S) where S ={A1,A2,A3} with A1= [1 0], A2= [0 0] and A3= [0 1]
.......................................................[0 0].......[0 1]...........[1 0]

part a) Identify W in more familiar terms.
My Answer: subspace

part b) Prove that S is a basis of W
My Answer: basis => linearly independent + span
span => Given in question
linearly independent => prove
therefore ?1[1 0] + ?2[0 0] + ?3[0 1] - [0 0]
................[0 0]..... [0 1] .....[1 0]....[0 0]

[?1 ?3] = [0 0]
[?3 ?2]......[0 0]

=> the scalars ?1, ?2 and ?3 are all zero => by definition, linearly independent

part c) Find the dimension of W
My Answer: dim(w) = 4

part d) Find one or more additional matrices A4,A5,... such that
T={A1,A2,A3,A4,A5...} is a basis of M22. justify your answer.
My Answer: A4= [0 0]
...................[0 0]
Still, scalars ?1, ?2, ?3, ?4 are all zero, therefore linearly independent
basis of M2x2

parte) For the marix X= [2 3], find the cooddinate vectors [X]S and [X]T.
............................[3 4]
My Answer: [X]s
x1[1 0] + x2[0 0] + x3[0 1]
...[0 0]....[0 1] ......[1 0]
=[2 3]
..[3 4] Hence x1=2 x3=3 and x2=4
=> [2]
....[4]
....[3]

[X]T
x1[1 0] + x2[0 0] + x3[0 1] + x4[0 0]
...[0 0]....[0 1].......[1 0]......[0 0]
= [2 3]
..[3 4] Hence x1=2 x3=3, x2=4 and x4=0
=[2]
..[4]
..[3]
..[0]


Plesae look through my answers and correct if im wrong
Thanks!!!!!