I just want to make sure my reasoning is correct.
Let A be a 7X3 matrix whose rank is 3
a)are the rows of A linearly dependent or independent?
I said dependent because a matrix whose rank is 3 means the row rank is 3, but if there are 7 rows then they have to be dependent on each other
b)are the columns of A linearly dependent or independent?
I said independent because the column rank would be 3 and there are 3 columns
is this reasoning enough to justify my answer?
Your reasoning is mostly correct, but let me explain in more detail to help reinforce your understanding.
a) To determine if the rows of A are linearly dependent or independent, one can examine the row rank. The row rank of A is the maximum number of linearly independent rows in the matrix.
In this case, you correctly stated that the rank of A is 3. Since the rank is the same as the row rank, it means there are 3 linearly independent rows in A. However, it does not necessarily mean that all 7 rows of A are dependent on each other. The remaining 4 rows might be linearly dependent on the first 3 rows, which would make the rows of A linearly dependent. To conclusively determine if the rows are dependent or independent, you would need more information about the specific matrix A.
b) Similarly, to determine if the columns of A are linearly dependent or independent, you can examine the column rank. The column rank is the maximum number of linearly independent columns in the matrix.
You correctly stated that the column rank of A is also 3, which means there are 3 linearly independent columns in A. Therefore, the columns of A are linearly independent. The fact that there are 3 columns, and the column rank is also 3, supports this conclusion.
In summary, your reasoning is mostly correct, but be careful when making assumptions about the row and column dependence solely based on the rank of the matrix. More information about the specific matrix is needed to determine the exact relationship between the rows or columns.