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.
Suppose is an -by- matrix, with rows and columns . The row space is the subspace of spanned by the vectors ; similarly, the column space is the subspace of spanned by the vectors .
Consider the matrix
Compute the row space and the column space .
The rows of this matrix are and , so the row space is the span of these two vectors in . In particular, since these rows are linearly independent, is a two-dimensional subspace of . To determine this subspace explicitly, note that both rows are orthogonal to , so they span the plane .
To determine the column space of , first note the columns of the matrix are , , and . Since the first two of these vectors are linearly independent, it follows that their span is a two-dimensional subspace of , and hence itself.
Note that, in the above computation, we have . This is an instance of the linear algebra fact that "row rank equals column rank," and is discussed in the article on rank.
Let be an -by- matrix, which corresponds to a linear transformation . One can interpret the row and column spaces of in terms of this transformation.
Suppose the columns of are . For any vector , one may compute This computation implies the image of is precisely the column space . One should think of this as a "coordinate-free" interpretation of the column space; no matter which matrix one chooses to represent the linear transformation , the column space of that matrix will always equal the image of .
If is a matrix representing a linear transformation , then the column space of is the image of . In symbols,
Similarly, one can interpret the row space of as follows: If the rows of are , then for any , one computes where denotes the dot product. Recall that the kernel of is the subspace of consisting of all the vectors such that . Then, the computation above shows that is in the kernel of if and only if it is orthogonal to each row . In other words, the kernel of is precisely the space of vectors such that for all .
Let be a vector space and a subspace; assume that has a dot (inner) product defined on it. The orthogonal complement of is the subspace consisting of the vectors such that for all .
If is a matrix representing a linear transformation , then the kernel of is the orthogonal complement of . In symbols,