%% Section 4.2 and Matlab

% Matlab note:  Transpose is denoted by '

%% 38.  Determine whether w is in the column space of A, the null space of A or both, where
w=[1 2 1 0]';
A=[-8 5 -2 0
    -5 2 1 -2
    10 -8 6 -3
    3 -2 1 0];

%% 39.  Let a1.. a5 denote the columns of A below.
A=[5 1 2 2 0
    3 3 2 -1 -12
    8 4 4 -5 12
    2 1 1 0 -2];
B=A(:,[1,2,4]); % B is made from cols 1, 2, 4 of A.
    
% a)  Explain why a3 and a5 are in the column space of B
% b)  Find a set of vectors that spans Null(A)
% c)  Let T(x)=Ax.  Explain why T is neither 1-1 nor onto.

%% 40.  Let H=span(v1, v2) and K=span(v3,v4) where v1..v4 are given below.
%  Then H and K are subspaces of R^3.  In fact, H and K are planes in R^3
%  that intersect in a line through the origin.  Find a vector w that
%  generates that line.  (Hint:  w is in both H and K-  Does that lead to a
%  system of equations that can be solved?)

v1=[5 3 8]';
v2=[1,3,4]';
v3=[2 -1 5]';
v4=[0 -12 -28]';