This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: MATRIX SPACES JOS ´ E MALAG ´ ONL ´ OPEZ The idea of this lecture is to give a name to the spaces and numbers that have been introduced before. This will give us a better understanding of our basic objects, matrices. Rank of a Matrix Recall that Gaussian elimination is given in two phases: given a matrix A , by means of elementary row operations we get two matrices A ∼ A ′ ∼ A ′′ , where A ′ is in REF, and A ′′ is in RREF. Remark 1 . Remember that the leading ones of the REF A ′ are exactly the same leading ones of the RREF A ′′ . In other words, Number of nonzero rows in A ′ = Number of leading ones in A ′ = Number of leading ones in A ′′ = Number of nonzero rows in A ′′ . Sometimes the leading ones in either A ′ or A ′′ are called pivots . Definition 2. Let A be a matrix. The rank of A , denoted as rank( A ), is the number of leading ones in A . Example 3. Since A = 3 2 − 2 1 1 2 2 − 1 ∼ 1 2 − 1 0 1 − 1 0 0 1 ∼ 1 0 0 0 1 0 0 0 1 we have that rank( A ) = 3. 1 Example 4. Since A = 0 1 2 1 2 3 2 3 4 ∼ 1 2 3 0 1 2 0 0 0 ∼ 1 0 − 1 0 1 2 0 0 we have that rank( A ) = 2. The following facts are a direct consequence of the definition of rank. Fact 5. Let A be any matrix. Then (1) rank( A ) ≤ number of rows in A . (2) rank( A ) ≤ number of columns in A . (3) A system of linear equations Avectorx = vector b is consistent if and only if rank( A ) = rank( A  vector b ) . As an example, consider the system ( A  vector b ) = 1 1 0 − 1 1 0 0 1 2 1 0 0 0 k In this case rank( A ) = 2 . Now, if k = 0 , we have that rank( A  vector b ) = 2 and the system is consistent. If k negationslash = 0 , we have that rank( A  vector b ) = 3 and the system is incon sistent. (4) The system Avectorx = vector b has infinitely many solutions if and only if rank( A  vector b ) = rank( A ) , and rank( A ) is less than the number of columns in A . (5) The system Avectorx = vector b has unique solution if and only if rank( A  vector b ) = rank( A ) = the number of columns in A. Matrix Spaces Let A be an m × n matrix. The columns of A , called the column vectors of A , are vectors in R m . The rows in A , called the row vectors of A , are vectors in R n . 2 Remark 6 . We have that (1) The system is consistent if and only if vector b is a linear combination of the column vectors of A . (2) The homogeneous system Avectorx = vector 0 has unique solution ( vectorx = vector 0), if and only if the column vectors of A are linearly independent in R m , if and only if the rank( A ) = n . The next is to give a name to the vector spaces related to the vectors mentioned above....
View
Full Document
 Fall '10
 JoseMalagonLopez
 Linear Algebra, Algebra, Matrices, Rank, Row

Click to edit the document details