1. Enter matrices A and b by typing 
and
.
a) Show that Ax=b is inconsistent.( Hint: you may show this by finding 
or finding both rank(A) and 
)
b) You can enter 
and A'*A by typing A'*A and A'* b in MATLAB. Show that the system 
is a consistent system and find a solution for 
c) Find the norm of the vector Ax-b by typing norm( Ax-b) where x is your solution from part (b).
i) Find a vector say , z, in column space of A by typing 
.
compute || z -b|| by typing norm( z -b), compare this value with || Ax-b|| which one is smaller?
ii) Find two other vectors, z, in cloumn space of A as we found z in part (i), find the norm of ||Az -b|| for each of the points, z, and compare the results with || Ax-b||.
d) Is A^t A invertible?
You may check this by
i) Finding det( A^t A) ( is it non-zero?) [Note : type det( A'* A)]
ii) Finding rref( A^t A) ( Is it an identity matrix?)[Note : type rref( A'* A)]
e) Are the columns of A linearly independent?
f) Are the rows of A are linearly independent?
g) Find a basis for the column space of A.
h) Find a basis for the row space of A.
i) How many solutions does Ax=0 have?
j) What is the number of solutions of A^t A x=0?
k) Find a basis for the range of the linear transformation T_A? ( T_A is defined by T(x)= Ax)
l) Find a basis for the null(T_A).
m) Find nullity of A , T_A and A^tA.
n) Find rank(A), rank(A^t), rank(T_A)and rank(A^t A)
v_1=( 1, 3, -2, 0, 2)
v_2=( 2, 6, -5, -2, 4)
v_3= (0, 0, 5, 10, 0)
v_4 =( 2, 6, 0, 8, 4)
(Note: form a matrix A whose columns are the vectors v_1, v_2, v_3, v_4, then find det(A^tA ).
b) Determine if the following vectors are linearly independent.
v_1=( 0, 0, -2, 0, 7, 12)
v_2=( 2, 4, -10, 6, 12, 28)
v_3= ( 2, 4, -5, 6, -5, -1)
3. Find an equation of the line that passes through the points ( 3,4) and (1,2).
a) Using the least square solution.
b) Using the regular way: finding slope and using slope intercept formula.
c) Did you get the same answer?
4. Consider the following set of points
(3, 4), (1,2), (-1, 1), (6,5), (7,9)
You may enter this points in a matrix by typing
D=[ 3 1 -1 6 7; 4 2 1 5 9; 1 1 1 1 1 ]'
then X= D(:,1) is the matrix of x-values and Y=D(:,2) is the matrix of y-values.
i) Find a polynomial of degree two y= a_2 x^2 +a_1 x + a_0 
, that best fits the points given above.