By Lin I-hsiung

This available publication for novices makes use of intuitive geometric innovations to create summary algebraic concept with a distinct emphasis on geometric characterizations. The publication applies identified effects to explain quite a few geometries and their invariants, and offers difficulties all for linear algebra, similar to in actual and complicated research, differential equations, differentiable manifolds, differential geometry, Markov chains and transformation teams. The transparent and inductive process makes this booklet detailed between latest books on linear algebra either in presentation and in content material.

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In particular, [O]B = 0, [X]B = 1. 2 Coordinatization of a Straight Line: R1 (or R) For example: 11 − [P ]B = −2 ⇔ OP = −2 x ; − 3 3 [Q]B = ⇔ OQ = x . 2 2 See Fig. 6. O P –2x X Q x 3x 2 L(O; X) Fig. 6 Now we summarize as The coordinatization of a straight line Let L(O; X) be an arbitrary vectorized space of line L, with B = { x }, −− x = OX, as a basis. The set RL(O;X) = {[P ]B | P ∈ L} is called the coordinatized space of L with respect to B. Explain further as follows. (1) There is a one-to-one correspondence from any point P on line L onto corresponding number [P ]B of the real number system R.

2. 2). For any three vectors b1 , b2 , b3 ∈ Σ(O; A1 , A2 ), prove that they are linearly dependent. 3. 4) in detail. 3 Coordinatization of a Plane: R2 Provide the plane Σ a ﬁxed vectorized space Σ(O; A1 , A2 ) with basis B = { a1 , a2 }, − − where a1 = OA1 and a2 = OA2 are called basis vectors. It should be mentioned that the order of appearance of a1 and a2 in B cannot be altered arbitrarily. Sometimes, it is just called an ordered basis to emphasize the order of the basis vectors. Hence, B is diﬀerent from the ordered basis { a2 , a1 }.

The direction from P to Q (along L)” is the same as “the direction from P to Q ”. 1) We call properties 1 and 2 as the parallel invariance of vector (see Fig. 4). P Q Fig. 4 P' Q' L The One-Dimensional Real Vector Space R (or R1 ) 8 In particular, for any points P and Q on L, one has − − P P = QQ = 0 . 2) Hence, zero vector is uniquely deﬁned. Now, ﬁx any two diﬀerent points O and X on L. e. −− x = OX. Note that x = 0 . − For any ﬁxed point P on L, the ratio of the signed length OP with −− respect to OX is − OP −− = α, OX where the real number α has the following properties: α = 0⇔P 0 < α < 1⇔P α > 1⇔P α = 1⇔P α < 0⇔P = O; lies on the segment OX (P = O, X); and X lie on the same side of O and OP > OX; = X; and and X lie on the diﬀerent sides of O.