What exactly do we mean when we are talking about the span o

What exactly do we mean when we are talking about the span of a matrix? (I understand the meaning of the span of a set of vectors)

Are we talking about the span of a column vectors of a matrix or the span of the row vectors of a matrix? Or are we talking about both the span of the column vectors and the span of row vectors and both spans happen to be equal to each other?

Please provide an intuitive explantion, not a mathematical one. Thank you.

Solution

Definition of spanning: A set of vectors spans a space if their linear combinations fill the space.

But what is the intuitive meaning of this, and the idea of a vector span? It is the process of solving by putting a matrix into reduced row-echelon form. In linear algebra, the column space C(A) of a matrix A (sometimes called the range of a matrix) is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image or range of the corresponding matrix transformation. The row space and column space of an m-by-n matrix are the linear subspaces generated by row vectors and column vectors, respectively, of the matrix. Its dimension is equal to the rank of the matrix and is at most min(m,n)

The column space is the other important vector space used in studying an m x n matrix. If we consider multiplication by a matrix as a sort of transformation that the vectors undergo, then the null space and the column space are the two natural collections of vectors which need to be studied to understand how this transformation works. While the null space focused on those vectors which vanished under action of the matrix (i.e. the solutions of Ax = 0) the column space corresponds to the transformed vectors themselves (i.e. all Ax). It is the totality of all the vectors after the transformation. Those readers who have studied abstract algebra can relate the null space to the kernel of a homomorphism and the column space to the range.

Another important space associated with the matrix is the row space. Like its name suggests it is built entirely out of the rows of the matrix. We shall later see that the row space can be identified with the column space in a particular sense. In the special case of an invertible matrix, the row space and the column space are exactly equal.