of its corresponding matrix has only the zero vector in its null space. Equivalently, a linear transformation is 1-to-1 if and only if its corresponding matrix has no non-pivot columns. • A field is an algebraic structure with addition, '+', and multiplication, '·', (and subtraction and division) with certain rules. A A homogeneous system of linear equations is a system of linear equations without constant terms. A homogeneous matrix vector equation has form Ax=0. The null space of the matrix A=[1,0,0,1] (the 2x2 identity matrix) is only the zero vector: null(A) = {0}, so it has nullity 0. ] is the set of all constant polynomials. The dimension of the space of all constant polynomials is 1, so the nullity of of its corresponding matrix has only the zero vector in its null space. Equivalently, a linear transformation is 1-to-1 if and only if its corresponding matrix has no non-pivot columns. is the leading non-zero element in each row. (We also refer to the pivots of a matrix which has not been reduced, referring implicitly to its echelon form.) , the act of pivoting is to perform a row swap (partial pivoting) or both a row and a column swap (total pivoting), in order to bring as large an element as possible into the pivot position. This can reduce accumulated error when performing numerical operations on large matrices. . The specification of a vector space includes specifying a field of scalars. The rules of scalar multiplication for that vector space apply to scalars from that field. C has complex numbers for scalars when viewed as a vector space over the complexes, but it can also be defined as a vector space with real scalars (a vector space 'over the reals') in which case a basis would be {1, Source.


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