Systems of Linear Equations We now examine the linear structure of the solution set to the linear system Ax = Thus the singular values of Aare ˙ 1 = 360 = 6 p 10, ˙ 2 = p 90 = 3 p 10, and ˙ 3 = 0. Further, from the foregoing row-echelon form of the augmented matrix of the system Ax = 0, we see that rank(A) = 2. We can also write it as . To compute the rank of a matrix, remember two key points: (i) the rank does not change under elementary row operations; (ii) the rank of a row-echelon matrix is easy to acquire. A matrix is an m×n array of scalars from a given field F. The individual values in the matrix are called entries. 2. THEOREM 1.3. We know that at least one of the eigenvalues is 0, because this matrix can have rank at most 2. proof. 3. rank(A) = m. This has important consequences. In fact, we can compute that the eigenvalues are p 1 = 360, 2 = 90, and 3 = 0. • has only the trivial solution . Dimensions of the row space and column space are equal for any matrix A. For example, is the normal form . The matrix rank algorithms can be readily applied to various problems in exact linear algebra, combinatorial optimization, and dynamic data structure. Hence, rank(A)+nullity(A) = 2 +2 = 4 = n, and the Rank-Nullity Theorem is verified. i.e. The row space of an m×n matrix A is the subspace of Rn spanned by rows of A. • The RREF of A is I. The dimension of the row space of A is called rank of A, and denoted rankA. We have seen that there exist an invertible m × m matrix Q and an invertible n × n matrix P such that A1 = Q−1AP has the block form I 0 A1 = 0 0 where I is an r × r identity matrix for some r, and the rest of the matrix is zero. • has a unique solution for all . [See the proof on p. 275 of the book.] Theorem. If order of matrix A is 3 x 3 2. Motivated by this, we convert the given matrix into row echelon form using elementary row operations: 2 6 6 4 0 16 8 4 2 4 8 16 16 8 4 2 4 8 16 2 3 7 7 5 ) 2 6 6 4 Let A be an n x n matrix. First, we show that the algorithms can be applied to computing a rank-one decomposition, finding a basis of the null space, and performing matrix multiplication for a low rank matrix. Chapter 2 Matrices and Linear Algebra 2.1 Basics Definition 2.1.1. Similarly is the normal form Definition: Rank and Nullity Theorem 2 If a matrix A is in row echelon form, then the nonzero rows of A are linearly independent. rank(A)=n,whereA is the matrix with columns v 1,...,v n. Fundamental Theorem of Invertible Matrices (extended) Theorem. Reducing it into the For this matrix, it is The following statements are equivalent: • A is invertible. Note : Rank of a Matrix is less than or equal to the least of its row or its column. The dimension of the row space is called the rank of the matrix A. Theorem 1 Elementary row operations do not change the row space of a matrix. Theorem 392 If A is an m n matrix, then the following statements are equivalent: 1. the system Ax = b is consistent for every m 1 matrix b. Hence rank of matrix A = 3. So, if m > n (more equations Definition : An m n matrix of rank r is said to be in normal form if it is of type. The row rank and the column rank of a matrix A are equal. So, rank of matrix B is 3. 1. Definition. The column space of A spans Rm. If order of matrix A is 2 x 3 Echelon Form Finding the rank of a matrix involves more computation work. If order of matrix A is 5 x 4 3. By above, the matrix in example 1 has rank 2. Recall, we saw earlier that if A is an m n matrix, then rank(A) min(m;n). 304-501 LINEAR SYSTEMS L5- 1/9 Lecture 7: Rank and Nullity of Matrices 2.6.4 Rank and Nullity of Matrices Let AU V: → be an LT, with dim{U}= n, dim{V}= m. This implies that A has an mn× matrix representation. Normal form of a Matrix We can find rank of a matrix by reducing it to normal form. By theorem, we could deflne rank as the dimension of the column space of A.

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