In each of the following, determine whether the indicated addition and scalar mu

Question

In each of the following, determine whether the indicated addition and scalar multiplication of ordered triples of real numbers yields a vector space. For those that are not vector spaces, determine which properties of a vector space fail to hold. (x1, y1, z1) + (x2, y2, z2) = (x1 + x2, y1 + y2, z1 + z2), c(x, y, z) = (cx, y, cz) (x1, y1, z1) + (x2, y2, z2) = (z1 +z2,y1 + y2, x1 + x2), c(x, y, z) = (cx, cy, cz) (x1, y1, z1) + (x2, y2, z2) = (x1 + x2, y1 + y2 - 2, z1 + z2), c(x, y, z) = (cx, y, z)

Explanation / Answer

Properties of Addition The basic properties of addition for real numbers also hold true for matrices. Let A, B and C be m x n matrices A + B = B + A commutative A + (B + C) = (A + B) + C associative There is a unique m x n matrix O with A + O = A additive identity For any m x n matrix A there is an m x n matrix B (called -A) with A + B = O additive inverse Properties of Matrix Multiplication Unlike matrix addition, the properties of multiplication of real numbers do not all generalize to matrices. Matrices rarely commute even if AB and BA are both defined. There often is no multiplicative inverse of a matrix, even if the matrix is a square matrix. There are a few properties of multiplication of real numbers that generalize to matrices. We state them now. Let A, B and C be matrices of dimensions such that the following are defined. Then A(BC) = (AB)C associative A(B + C) = AB + AC distributive (A + B)C = AC + BC distributive There are unique matrices Im and In with Im A = A In = A multiplicative identity We will often omit the subscript and write I for the identity matrix. The identity matrix is a square scalar matrix with 1's along the diagonal.

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