Heisenberg had not proceeded very far with this idea before he noticed that it would lead to his physical quantities not satisfying the commutative law of multiplication.

在意识到矩阵表示将导致物理量不满足乘法交换律之前，海森堡并没有前进太远。

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Basic operation rules have been established, including addition, positive scalar multiplication, cancellation law for addition and so on.

基本的运算法则已经形成，包括加法运算、数乘运算、加法的消去律等。

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Here we have used the associative law or the cancellation law for multiplication, which are satisfied for natural Numbers, now we assume they are satisfied with fraction Numbers.

这里我们用到了乘法结合律也可以用消去律，本来这些运算律是对正整数乘法适用的，但对于有分数参加的乘法我们也规定适用。

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