Just do the problem 5cSolutionfull rank A means it column ve

Just do the problem 5c

Solution

full rank A means: it\' column vectors a1, a2, an are linearly independent // note- A\' means A transpose

say, vector x belongs to N(A\'A) , therefore, A\'Ax=0

premultiply by v\' on both side , we get x\'A\'Ax=x\'0=0

then , lhs= (Ax)\'(Ax)=0

||Ax||² =0   

length of a vector is zero means , it\'s a zero vector therefore, Ax vector is zero // x is vector and A is a matrix of m by n

since x was the member of null space of (A\'A) // we asumed initially

then x is also the member of N(A) b/c we proved that Ax=0

Now, null space of A is 0 because it has full column rank =n , i.e. all columns are independent

now, x is member of N(A\'A) as well as N(A) and N(A) has only 0 vector therefore , x is also a zero vector

therefore , N(A\'A)=N(A)=0

therefore, only solution to (A\'A)x=0 is 0 vector , hence columns of A\'A are linearly independent b/c N(A\'A)=0

therefore rank of A\'A is n that is full column rank, and we know that if rank is full column rank then that matrix is invertible b/c |A\'A| is not equal to 0.