MATRIX THEORY-CHAPER 0 3.RANK Definition 4.Let Ve be a finite dimensional vector space.We call the vectorsv...xV linearly independent if the equation x1+22+…k,x∈ only has zero solution 1=2=...x=0. 1,e2, Definition 5.Let S be a set of vectors.The rank of S is defined by the marimum number r such that there erists r vectorsv,v2,..vS such that they are linearly independent. Example&aA-[名！g】emm到-2 Theorem 4.Let AE Mm.n(F).Then rank(A)=rank(AT)=rank(A')<min{m,n} Definition 7.The following operations are called elementary operation .Interchange of two rows; ·AMh tion of a row by a nonzero scalar; Definition 8.The RREF is the matrices satisfying 包Au er the时chm ro row has 1 as its first no Example 4.Find the RREF for the following matriz 「1 A- Then rank(A)=2.Moreover, Nul(A)={z∈C:Az=0 Ran(A)={y∈c4:y=Ax}= Theorem 6.Given any then dim(Null(A))=n-rank(A),dim(Ran(A))=rank(A). MATRIX THEORY - CHAPER 0 3 3. Rank Definition 4. Let VF be a finite dimensional vector space. We call the vectors v1, v2, · · · , vk ∈ V linearly independent if the equation x1v1 + x2v2 + · · · xkvk, xi ∈ F. only has zero solution x1 = x2 = · · · xk = 0. Example 2. Given any vectors space V of finite dimension n. If {e1, e2, · · · , en} is a basis of V , then e1, e2, · · · , en are linearly independent. Definition 5. Let S be a set of vectors. The rank of S is defined by the maximum number r such that there exists r vectors v1, v2, · · · , vr ∈ S such that they are linearly independent. Definition 6. Let A ∈ Mn,m(F). Then the rank of A is defined by the rank of column vectors, or equivalently the rank of row vectors. Example 3. Let A =  2 0 3 −1 4 5  . Then rank(A) = 2. Theorem 4. Let A ∈ Mm,n(F). Then rank(A) = rank(A T ) = rank(A ∗ ) ≤ min{m, n} Definition 7. The following operations are called elementary operation: • Interchange of two rows; • Multiplication of a row by a nonzero scalar; • Addition of a scalar multiple of one two to another row. Theorem 5. Any matrix A ∈ Mm,n(F) corresponds to a canonical form , called row-reduced echelon form (RREF), by a sequence of elementary operations. Definition 8. The RREF is the matrices satisfying (a) Each nonzero row has 1 as its first nonzero entry; (b) All other entries in the column of such a leading 1 are 0; (c) Any rows consisting entirely of zeroes occur at the bottom of the matrix; (d) The leading 1’s occur in a stair step pattern, left to right. Example 4. Find the RREF for the following matrix: A =     1 0 2 1 1 −1 2 1 3 −1 1 2 5 5 1 2 −2 1 −2 2     −→     1 0 2 1 1 0 2 3 4 0 0 2 3 4 0 0 −2 −3 −4 0     −→     1 0 2 1 1 0 1 3 2 2 0 0 0 0 0 0 0 0 0 0 0     Then rank(A) = 2. Moreover, Null(A) = {x ∈ C 5 : Ax = 0} = Span      −1 0 0 0 1   ,   −1 −2 0 1 0   ,   0 − 3 2 1 0 0      , Ran(A) = {y ∈ C 4 : y = Ax} = Span      1 −1 1 2   ,   0 2 2 −2      . Theorem 6. Given any A ∈ Mm,n, then dim(Null(A)) = n − rank(A), dim(Ran(A)) = rank(A)