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.

How do you denote row space?

The space spanned by the rows of A is called the row space of A, denoted RS(A); it is a subspace of R n . The space spanned by the columns of A is called the column space of A, denoted CS(A); it is a subspace of R m .

What is basis for row space?

2. A basis for the row space of a matrix A consists of the row vectors of any row-echelon form of A. 3. The nonzero column vectors of a row-echelon form of a matrix A form a basis for colspace(A).

How do you write row space?

Linear Algebra The space spanned by the rows of A is called the row space of A, denoted RS(A); it is a subspace of R n . The space spanned by the columns of A is called the column space of A, denoted CS(A); it is a subspace of R m .

What is Col A linear algebra?

Definition: The Column Space of a matrix “A” is the set “Col A “of all linear combinations of the columns of “A”.

Is the row space a 3 dimensional subspace?

Note that since the row space is a 3‐dimensional subspace of R 3, it must be all of R 3. Criteria for membership in the column space. If A is an m x n matrix and x is an n ‐vector, written as a column matrix, then the product A x is equal to a linear combination of the columns of A :

Is there a basis for the row space of Rs?

The collection { r 1, r 2, …, r m } consisting of the rows of A may not form a basis for RS (A), because the collection may not be linearly independent. However, a maximal linearly independent subset of { r 1, r 2, …, r m } does give a basis for the row space.

What is the row space and column space of a matrix?

Row Space and Column Space of a Matrix. Let A be an m by n matrix. The space spanned by the rows of A is called the row space of A, denoted RS(A); it is a subspace of R n . The space spanned by the columns of A is called the column space of A, denoted CS(A); it is a subspace of R m .

How do you find the basis of a column space?

Since the column space of A consists precisely of those vectors b such that A x = b is a solvable system, one way to determine a basis for CS (A) would be to first find the space of all vectors b such that A x = b is consistent, then constructing a basis for this space.