Linear Algebra AX=B

Question

Consider a matrix A (n×m) ,X(m×n) and B(n×n) such that AX=B . If A has k linearly independent columns then what conclusions can we nake about the number of linearly independent columns of B.

Answer 1

Let's consider a matrix \( A \) of size \( n \times m \) with \( k \) linearly independent columns.

Suppose \( A \) has the following form:

\[ A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{pmatrix} \]

This matrix has \( k = 2 \) linearly independent columns.

Now, let's assume \( X \) is a matrix of size \( m \times n \), such that the multiplication \( AX \) is defined and results in a matrix \( B \) of size \( n \times n \).

If \( AX = B \), then the number of columns in \( B \) will be the same as the number of columns in \( X \), which is \( n \).

So, in our case, \( B \) will be a \( n \times n \) matrix.

Now, the rank of \( B \) will be at most \( k \) (the rank of \( A \)).

Let's take an example where \( B \) has a rank equal to \( k \):

\[ B = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix} \]

Here, \( B \) has \( k = 2 \) linearly independent columns.

In summary, if \( A \) has \( k \) linearly independent columns, then \( B \) can have at most \( k \) linearly independent columns, but it may have fewer.