What does the row space of a matrix represent?

If you think of the rows of matrix A as vectors, then the row space is the set of all vectors that are linear combinations of the rows. In other words, it is the set of all vectors y such that ATx=y for some vector x.

Why is row space and column space important?

In linear algebra, when studying a particular matrix, one is often interested in determining vector spaces associated with the matrix, so as to better understand how the corresponding linear transformation operates. Two important examples of associated subspaces are the row space and column space of a matrix.

What is the importance of rank of a matrix?

In control theory, the rank of a matrix can be used to determine whether a linear system is controllable, or observable. In the field of communication complexity, the rank of the communication matrix of a function gives bounds on the amount of communication needed for two parties to compute the function.

What is the meaning of row space?

The vector space generated by the rows of a matrix viewed as vectors. The row space of a matrix with real entries is a subspace generated by elements of , hence its dimension is at most equal to . It is equal to the dimension of the column space of (as will be shown below), and is called the rank of .

What is basis of row space?

The nonzero rows of a matrix in reduced row echelon form are clearly independent and therefore will always form a basis for the row space of A. Thus the dimension of the row space of A is the number of leading 1’s in rref(A). Theorem: The row space of A is equal to the row space of rref(A).

Why is column space important?

An important property: The linear system Ax=b has a solution if and only if b belongs to the column space of A. Since linear systems of equations arise often in practice (particularly when working with computers), knowing when a linear system of equations has a solution may be very useful.

What is the use of row space?

Given what we know about spans and matrices, the row space is just the span of each of the rows, if we are to consider each row to be a vector in a set. Recall that the span is just the set of all linear combinations of a set of vectors, which describes the space that is reachable by those linear combinations.

Why row rank is equal to column rank?

THEOREM: If A is an m×n matrix, then the row rank of A is equal to the column rank of A. Thus the r rows of C form a minimal spanning set of the row space of A and the r columns of B form a minimal spanning set of the column space of A. Hence, row and column ranks are both r.

What row rank means?

The row rank of a matrix is the maximum number of rows, thought of as vectors, which are linearly independent. Similarly, the column rank is the maximum number of columns which are linearly indepen- dent. It is an important result, not too hard to show that the row and column ranks of a matrix are equal to each other.