Thread: Question on null space/column space/row space of a matrix

Could someone please tell me how to answer the questions a, b and c? I really don't know where to start, since the matrix P is not given... Thanks for your help!

B =

Let P be any invertible 3 x 3 matrix and define the matrix C as: C = PB

(a) Considering the relationship between B and C, what can you say about Nul(C)?
(b) What about Col(C)?
(c) What about Row(C)?

In answering the above questions, note that your statements must be valid for all invertible 3 x 3 matrices P (and not just for a particular choice of P). You should address the dimensions of the spaces and whenever possible you should say something about the spaces themselves, comparing those of C to those of B.

The null space of an invertible 3x3 matrix is the 0 matrix, because invertibility implies that applying a finite sequence of elementary row operations to a matrix eventually yields the identity matrix. What is the solution space to the homogeneous system Ax = 0, where A is any invertible matrix? Since A is row and column equivalent to the identity matrix, isn't it the zero vector? Also, rank-nullity theorem: rank + nullity = n, where n is the number of columns. If the matrix is invertible, then rank is equal to the number of columns and nullity = 0.

As for the column space, it is simply the span of all of the columns of an invertible matrix. Why? Because the rank for an invertible matrix is equal to the number of columns by the rank-nullity theorem. Intuitively, none of the columns disappear (linearly dependence) for an invertible matrix; all of the columns are linearly independent and even form a basis for the column space.

This can be generalized to the row space as well.

Thanks, but your answer only applies to the matrix P. What about the matrix C?

You appear to be completely misunderstanding the question. You say "the matrix P is not given". P is to be any 3 by 3 matrix.

I think I understand now. This is what I got.

B =

Let P be any invertible 3 x 3 matrix and define the matrix C as: C = PB

(a) Considering the relationship between B and C, what can you say about Nul(C)?

The dimension of the column space of matrix B equals 2 (2 pivot columns), so the rank of matrix B equals 2.
Multiplying a matrix by an invertible matrix does not change its rank, so the rank of matrix C equals 2.
Because the sum of the rank and nullity of a matrix equals the number of columns, the nullity of matrix C equals 4 - 2 = 2 (matrix C has 4 columns).
This means the dimension of the null space of matrix C equals 2.

(b) What about Col(C)?

The rank of matrix C equals 2, so the dimension of the column space of matrix C equals 2.

(c) What about Row(C)?

The rank of matrix C equals 2, so the dimension of the row space of matrix C equals 2.

So I found an answer for the dimensions of the spaces. But what about the spaces themselves? The question was:

In answering the above questions, note that your statements must be valid for all invertible 3 x 3 matrices P (and not just for a particular choice of P). You should address the dimensions of the spaces and whenever possible you should say something about the spaces themselves, comparing those of C to those of B.

What can you say about the spaces themselves?

Please help!

Is it true that:

Nul(B) = Nul(C)?
Col(B) = Col(C)?
Row(B) = Row(C)?

Why/why not?

If this is not what is meant by "whenever possible you should say something about the spaces themselves, comparing those of C to those of B.", then what is?