Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 sage: A = MatrixSpace(IntegerRing(), 2)([1,2,3,4]) sage: A.row_module() Free module of degree 2 and rank 2 over Integer Ring Echelon basis matrix: [1 0] [0 2] row_space(base_ring=None) Return the row space of this matrix. (Synonym for self.row_module().) EXAMPLES: sage: t = matrix(QQ, 3, range(9)); t [0 1 2] [3 4 5] [6 7 8] sage: t.row_space() Vector space of degree 3 and dimension 2 over Rational Field Basis matrix: [ 1 0 -1] [ 0 1 2] sage: m = Matrix(Integers(5),2,2,[2,2,2,2]); sage: m.row_space() Vector space of degree 2 and dimension 1 over Ring of integers modulo 5 Basis matrix: [1 1] rref(*args, **kwds) Return the reduced row echelon form of the matrix, considered as a matrix over a field. If the matrix is over a ring, then an equivalent matrix is constructed over the fraction field, and then row reduced. All arguments are passed on to :meth:echelon_form. Note: Because the matrix is viewed as a matrix over a field, every leading coefficient of the returned matrix will be one and will be the only nonzero entry in its column. EXAMPLES: sage: A=matrix(3,range(9)); A [0 1 2] [3 4 5] [6 7 8] sage: A.rref() [ 1 0 -1] [ 0 1 2] [ 0 0 0] Note that there is a difference between rref() and echelon_form() when the matrix is not over a field (in this case, the integers instead of the rational numbers): sage: A.base_ring() Integer Ring sage: A.echelon_form() [ 3 0 -3] [ 0 1 2] [ 0 0 0] 250 Chapter 7. Base class for matrices, part 2
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 sage: B=random_matrix(QQ,3,num_bound=10); B [ -4 -3 6] [ 5 -5 9/2] [3/2 -4 -7] sage: B.rref() [1 0 0] [0 1 0] [0 0 1] In this case, since B is a matrix over a field (the rational numbers), rref() and echelon_form() are exactly the same: sage: B.echelon_form() [1 0 0] [0 1 0] [0 0 1] sage: B.echelon_form() is B.rref() True Since echelon_form() is not implemented for every ring, sometimes behavior varies, as here: sage: R.=ZZ[] sage: C = matrix(3,[2,x,x^2,x+1,3-x,-1,3,2,1]) sage: C.rref() [1 0 0] [0 1 0] [0 0 1] sage: C.base_ring() Univariate Polynomial Ring in x over Integer Ring sage: C.echelon_form() Traceback (most recent call last): ... NotImplementedError: Ideal Ideal (2, x + 1) of Univariate Polynomial Ring in x over Integer Echelon form not implemented over ’Univariate Polynomial Ring in x over Integer Ring’. sage: C = matrix(3,[2,x,x^2,x+1,3-x,-1,3,2,1/2]) sage: C.echelon_form() [ 2 x [ 0 1 15*x^2 - 3/2*x [ 0 0 5/2*x^3 - 15/4*x^2 - 9/4* sage: C.rref() [1 0 0] [0 1 0] [0 0 1] sage: C = matrix(3,[2,x,x^2,x+1,3-x,-1/x,3,2,1/2]) sage: C.echelon_form() [1 0 0] [0 1 0] [0 0 1] set_block(row, col, block) Sets the sub-matrix of self, with upper left corner given by row, col to block. EXAMPLES: sage: A = matrix(QQ, 3, 3, range(9))/2 sage: B = matrix(ZZ, 2, 1, [100,200]) sage: A.set_block(0, 1, B) sage: A [ 0 100 1] 251
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Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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