By Robin Chapman
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Additional info for A Guide to Arithmetic [Lecture notes]
X1, x2) + ( y1, y2) OP + OQ ( y1, y2) OP OQ a2 O OP Φ a1 (x1, x2) (0, 1) (0, 0) Σ(O; A1, A2) (x1, x2) (1, 0) R2Σ(O; A1, A2) Fig. 16 The Two-Dimensional Real Vector Space R2 36 Notice that the linear isomorphism Φ carries overall vector operations in Σ(O; A1 , A2 ) to the corresponding vector operations in R2Σ(O; A1 , A2 ) , a neat, easy-to-handle, but abstract space. In particular, Φ preserves the linear dependence and independence of vectors. To see these, suppose b1 = α11 a1 + α12 a2 and b2 = α21 a1 + α22 a2 are in Σ(O; A1 , A2 ), then Φ(y1 b1 + y2 b2 ) = Φ((y1 α11 + y2 α21 ) a1 + (y1 α12 + y2 α22 ) a2 ) = (y1 α11 + y2 α21 , y1 α12 + y2 α22 ) = y1 (α11 , α12 ) + y2 (α21 , α22 ) = y1 Φ( b1 ) + y2 Φ( b2 ) Since Φ is one-to-one and onto, what we claimed follows easily.
P Q = RS (in length), and 2. the direction from P to Q is the same as the direction from R to S. 3) This is called the parallel invariance of vectors. As indicated in Fig. 6, − − P R = QS. S R Q P Fig. 6 26 The Two-Dimensional Real Vector Space R2 Usually, we use x to represent a vector, that is to say, for any ﬁxed point P , one can always ﬁnd another point Q such that − x = P Q. − According to parallel invariance, there are inﬁnitely many choices of P Q to − represent the same x , once these P Q have the same length and the same direction.
5). 5) further. For this purpose, deﬁne mappings See Fig. 18. Φ1 : R → RL(O; A1 ) by Φ1 (x) = (x, 0), Φ2 : R → RL(O; A2 ) by Φ2 (x) = (0, x). 8) The Two-Dimensional Real Vector Space R2 40 Φ1 R x 0 RL(O; A2) Φ 1(x) = (x, 0) A2 e2 O 1 0 Φ2 e1 Φ 2(x) = (0, x) A1 RL(O; A1) Fig. e. one-to-one, onto and preserving two vector operations) between vector spaces concerned. 6). 7)) R2 = R ⊕ R, the (external) direct sum of its subspace R. Exercises 1. 3), prove that Ψ−1 ◦Φ: Σ(O; A1 , A2 ) → Σ(O ; B1 , B2 ) is a linear isomorphism.