Linear Algebra/Row and column spaces

Having gone over vector spaces, we now combine the ideas of bases and dimension with the vector spaces, and the ideas of linear dependence and rank applied earlier. We will apply these important ideas to the general solution of m linear equations of n variables.

Definitions

Suppose we have an a field F and an m×n matrix of elements of F, and let r be the rank of that matrix.

Consider the columns or rows of the matrix are elements of a vector space where addition and scalar multiplication were defined before. One can easily check that they form a vector space.

Let M be a minor of the matrix F with order r. This minor is called the basis minor, and the columns and rows of this minor are called the basis columns and basis rows. The column space and row space of a matrix are respectively the vector spaces spanned by the columns and rows of the matrix.

Theorem

The dimension of both the basis and column spaces of a matrix are both equal to r, the rank of the matrix, and the basis columns (or rows) form a basis for the column (or row) space.

Proof

Let 𝐴 be a 𝑚×𝑛 matrix with rank 𝑟. Consider a matrix 𝐿 𝑚×𝑟 which is formed by the 𝑟 columns of 𝐴 forming the bases (vectors). By construction of 𝐿, each column of 𝐴 can be written as a linear combination of the columns of 𝐿 i.e. there exist a matrix 𝑍 𝑟×𝑛 such that 𝐴=𝐿𝑍. This means that each row of 𝐴 can be written as a linear combination of the 𝑟 rows of 𝑍 i.e. the row space of 𝐴is spanned by the rows of 𝑍. That is, row rank of 𝐴 is bounded above by 𝑟. Also, 𝐴^𝑇=𝑍^𝑇𝐿^𝑇. Clearly, the range of 𝑍^𝑇𝐿^𝑇 is contained in the range of 𝑍^𝑇. So, the rank of 𝐴^𝑇 is at most the rank of 𝐴. This argument holds for any matrix. Using the above argument and a similar argument starting with 𝐴𝑇, we conclude that transposition does not change the rank of the matrix.

Corollaries

1. Any n columns are linearly dependent when n>r.
2. The columns (or rows) of a matrix are linearly dependent when the number of columns (or rows) is greater than the rank, and are linearly independent when the number of columns (or rows) is equal to the rank.
3. The maximum number of linearly independent rows equals the maximum number of linearly independent columns.