Is the rank of a matrix the same as the dimension?

The rank of a matrix is the dimension of the image of the linear transformation represented by the matrix. The image is the column space of the matrix, so the rank is the dimension of the column space, and consequently equal to the number of linearly independent columns.

How are the rank dimension of the null space and the size of a related?

Rank: Rank of a matrix refers to the number of linearly independent rows or columns of the matrix. The number of parameter in the general solution is the dimension of the null space (which is 1 in this example). Thus, the sum of the rank and the nullity of A is 2 + 1 which is equal to the number of columns of A.

How do you find the dimension of the solution space of a matrix?

Let A be an m by n matrix and let r be the rank of A. The solution space of the system of linear equations AX = O is a subspace of Vn of dimension n − r. In particular, if the rows of A are independent then the solution space of the system AX = O has dimension n − m. A proof is in [Mun, B, Theorem 3].

What is the dimension of a matrix equal to?

The dimensions of a matrix are the number of rows by the number of columns. If a matrix has a rows and b columns, it is an a×b matrix. For example, the first matrix shown below is a 2×2 matrix; the second one is a 1×4 matrix; and the third one is a 3×3 matrix.

What is meant by rank of a matrix?

The rank of a matrix is the maximum number of its linearly independent column vectors (or row vectors). From this definition it is obvious that the rank of a matrix cannot exceed the number of its rows (or columns).

What is the dimension of the solution space of the homogeneous system?

Corollary 8.4. The dimension of the solution space of an n × m homogeneous linear system is m − r where m is the (column) rank of the corresponding coefficient matrix.