Explain a There is no matrix whose row space contains 1 2 1

Explain. a) There is no matrix whose row space contains (1, 2, 1, 1) and whose nullspace contains (1, -2, 1, 1). b) If A R^mxn then a linear system Ax = b with rank(A) = r = m and r

Solution

a)False Since the two row space and null space are linearly independent.Any subspace(here null space) basis elements are part of whole vector space basis(here row space).

b)False It will have infinite solution .

Since in this case no of variables = n and rank is r (i.e. no of equations)

Hence In this case r<n i,e No of equation less than No of variables.

c)True

There are several ways to show this for the column vectors.  

Consider the matrix equation Ax=b and let A be invertible. This means that there is exactly one vector x that is a solution to Ax=b. Now b is a nx1 vector and A is an nxn matrix. Recall that this means that b is a linear combination of the column vectors of A. Since there is a solution for all vectors b this means that the column vectors of A span Rn and there are n of them. Therefore they must be linearly independent.

d) FALSE

The sum of the dimensions of the row space and the null space of A equals the number of rows in A false but It is equals number of columns by rank theorem.

Also dimension of row space = number pivot columns, dimension of null space =number of non-pivot columns (free variables) so these add to total number of columns.

e) TRUE

Since the existence of multiple solutions (provided that there is any solution at all) depends only on the coecient matrix and since a homogeneous system always has at least one solution (namely the trivial one), multiple solutions for a linear system are possible only if the corresponding homogeneous system has multiple solutions. But the homogeneous system has multiple solutions if and only if it has a non-trivial solution.