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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....
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This note was uploaded on 01/22/2011 for the course MAT 1341 taught by Professor Josemalagonlopez during the Fall '10 term at University of Ottawa.
 Fall '10
 JoseMalagonLopez
 Linear Algebra, Algebra, Matrices

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