Ads
related to: linear algebra row rank questions examples worksheet 1 2 and 1 4 equal 3 4kutasoftware.com has been visited by 10K+ users in the past month
Search results
Results from the WOW.Com Content Network
In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns. [1] [2] [3] This corresponds to the maximal number of linearly independent columns of A. This, in turn, is identical to the dimension of the vector space spanned by its rows. [4]
Once the matrix is in echelon form, the nonzero rows are a basis for the row space. In this case, the basis is { [1, 3, 2], [2, 7, 4] }. Another possible basis { [1, 0, 2], [0, 1, 0] } comes from a further reduction. [9] This algorithm can be used in general to find a basis for the span of a set of vectors.
The rank of a matrix is equal to the dimension of the row space, so row equivalent matrices must have the same rank. This is equal to the number of pivots in the reduced row echelon form. A matrix is invertible if and only if it is row equivalent to the identity matrix.
Nonsingularity of the latter requires that B −1 exist since it equals B(I + VA −1 UB) and the rank of the latter cannot exceed the rank of B. [7] Since B is invertible, the two B terms flanking the parenthetical quantity inverse in the right-hand side can be replaced with (B −1) −1, which results in the original Woodbury identity.
For example, to solve a system of n equations for n unknowns by performing row operations on the matrix until it is in echelon form, and then solving for each unknown in reverse order, requires n(n + 1)/2 divisions, (2n 3 + 3n 2 − 5n)/6 multiplications, and (2n 3 + 3n 2 − 5n)/6 subtractions, [10] for a total of approximately 2n 3 /3 operations.
Consider the system of equations x + y + 2z = 3, x + y + z = 1, 2x + 2y + 2z = 2.. The coefficient matrix is = [], and the augmented matrix is (|) = [].Since both of these have the same rank, namely 2, there exists at least one solution; and since their rank is less than the number of unknowns, the latter being 3, there are infinitely many solutions.
A matrix is in reduced row echelon form if it is in row echelon form, with the additional property that the first nonzero entry of each row is equal to and is the only nonzero entry of its column. The reduced row echelon form of a matrix is unique and does not depend on the sequence of elementary row operations used to obtain it.
[1] [2] Variants of the latter have been shown to be weakly stable (i.e. they exhibit numerical stability for well-conditioned linear systems). [3] The algorithms can also be used to find the determinant of a Toeplitz matrix in O ( n 2 ) {\displaystyle O(n^{2})} time.
Ads
related to: linear algebra row rank questions examples worksheet 1 2 and 1 4 equal 3 4kutasoftware.com has been visited by 10K+ users in the past month