Let M be an m times n matrix of ranks s Find dim Nul M and d

Let M be an m times n matrix of ranks s. Find dim Nul (M) and dim (Nul (M^T). Let A = [1 -1 1 1 3 2 3 1 4] and b = [-1 0 4]. Find all the least square solutions to the system.

Solution

2.

Theorem: The rank of a matrix M plus the nullity of M equals the number of columns of M.

Now,

Rank of matrix M = s

Column of matrix M = n

So, the nullity of matrix M is n – s

So, dim Nul(M) = n – s

Similarly,

Rank of matrix M^T = s

Column of matrix M^T = m

So, the nullity of matrix M is m – s

So, dim Nul(M^T) = m – s