Q1:
Observation:
For every vector in my case, I compared rank and row of the vector. If they are equal then the result is
linear independent, else they are linear dependent.
I have created a matrix with a 10x10 and 10x9 scale for our observations. The result was linear
independent for every matrix. However, when the column is bigger according to case three. The result
became linear dependent.
1.
For case 1: I used 20x20 and make a random matrix. The result was a unique solutions x. Because the
rank from A and the augment matrix(A,b) is equal.
For case 2: I used 30x20. Because of the rank from A is smaller than the rank of (A, b), the result was
inconsistent.
For case 3: I used 20x30. The result was infinitely solution since the rank from A and (A,b) smaller than
the row of x.
Q2:
1.
S1= [1 2 -1 3 ; 4 1 1 8 ; 1 0 2 2]
The rref form of S1 is a linear independent so the subspace of s1 is not equal to R^4. The subspace of S1
is a hyperplane because the dimension of S1 is 2.
2.
As I row-reduce the matrix with Z2, I saw the last column does not contain all zero. So, z2 is not in S1
As for Z3, it had a solution set so Z3 is in S1.
3. The dimension is 2.
4.The dimension is 1