Matrix Rank Flashcards
What is the rank of a matrix?
The space spanned by the n column vectors of a matrix.
It is the maximum number of column vectors which are linearly independent.
What does a m*n matrix represent?
A linear mapping from a space of dimension n into a space of dimension m.
Is the rank of A equal to the rank of A^T?
Yes, so we can replace columns by rows.
What must rank be equal to or less than?
The number of columns/rows.
What happens when r < m (rows) ?
There may not be a solution.
What happens when r < n (number of coordinates of vector x) ?
There is not enough information to determine a unique solution.
What is the rank of a matrix when it is in echelon form?
The number of rows with not all zero numbers.
When will the linear system have a unique solution?
If r = n.
When will a system have infinitely many solutions?
If r < n (with at least 1 row of all zeros).
What happens when A and ^A do not have the same rank?
There are no solutions.
