Let A be a 5 by 4 matris and suppose that the system Ax=b has a solution. Explain why the solution is unique if and only if the rank of A is 4.

I said that it is unique becaue the rank is equal to the number of columns.

Do you have a better explaination.

Your reasoning is partially correct. The uniqueness of the solution to the system Ax=b is indeed related to the rank of matrix A, but there is another key factor to consider. Let's break down the explanation step by step:

1. The rank of a matrix A represents the maximum number of linearly independent rows or columns in A. In other words, it measures the dimension of the column space or the row space of A.

2. In the case of a 5 by 4 matrix A, if the rank is 4, it means that there are 4 linearly independent columns in A. This also implies that the column space (the space spanned by all the columns of A) has dimension 4.

3. Now, if the system Ax=b has a solution, it means that there exists at least one combination of the columns of A that can produce the vector b. Geometrically, b lies within the column space of A.

4. When the rank of A is 4, the column space has dimension 4, which means it fills the entire 4-dimensional space. Consequently, there is no room for any other linear combination of the columns of A to produce the same vector b. Therefore, when the rank is 4, the solution to Ax=b is unique.

5. On the other hand, if the rank of A is less than 4, for example, 3, it means that the column space has dimension 3, leaving one dimension unaccounted for. This implies that there might be more than one linear combination of the columns of A that yields the vector b. Thus, the system Ax=b would have multiple solutions, and the solution would not be unique.

In summary, the solution to the system Ax=b is unique if and only if the rank of A is equal to the number of columns, which in this case is 4.