By Spencer, Donald Clayton; Nickerson, Helen Kelsall

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**Example text**

12. 4. set Definition. A binary relation defined on a S is called an equivalence relation if it satisfies the following axioms: 20 Axiom 1. Reflexivity: Axiom Symmetry: 2• Axiom 3. 5. A = A · for ·each A e 1f s. then B = A. A= B and B = C, then c. e Proposition. defines a partition of An equivalence relation on a set S S into non-empty, mutually disjoint sub- sets called equivalence classes, such that two elements of S lie in the same equivalence class i f and only i f they are equivalent. The proof of this proposition is left as an exercise.

Rk, Find a basis for thus proving k = dim Rk . = o, the set of vectors in R3 of the form (x, 2X, 3X) for 5. (a) (b) all x (c) R, e RN where 6. N is a set of n If dim V such that L(D) 7, = V, and i f D is a set of k §12. of V of the form Show that the linear subspace + 1. is not finite dimensional. Parallels and affine subspaces Definition. A a vector of v. 1. vectors D is independent. For n = o, 1, ••• , is independent. ~ k, show that defined by fn(x) = xn. 11). v. X e u. The set A + U is It is said to result from parallel U by the vector A.

Is a linear subspace of is a linear subspace of (iii) If A1 , ••• , Ak are in V, then its image T(U) w. V, and x 1 , ••. , xk are in R, then T(E~=1xiAi) (iv) If D ( V, then L(D) and = E~=lxiT(Ai) is the linear subspace spanned by D, T(L(D)) = L(T(D)). Proof. of Definition 1 . 1 Since O'av ='av' the linearity property (2) gives This proves (i). To prove (ii), let x € R. Then A' and B' A', B1 be vectors in are the images of vectors Using the linearity property (1 ), we have A' + B 1 = T(A) + T(B) = T(A + B) • T(U ), A, B and of u.