4<\/a>\uff0c\u7cfb\u7edf\u9610\u91ca\u4e86\u4e3a\u4f55\u4e94\u6b21\u4ee5\u4e0a\u4e4b\u65b9\u7a0b\u5f0f\u6ca1\u6709\u516c\u5f0f\u89e3\uff0c\u800c\u56db\u6b21\u4ee5\u4e0b\u6709\u516c\u5f0f\u89e3[5]\uff0c\u4f7f\u4ee3\u6570\u5b66\u4ece\u89e3\u65b9\u7a0b\u7684\u79d1\u5b66\u8f6c\u53d8\u4e3a\u7814\u7a76\u4ee3\u6570\u7ed3\u6784\u7684\u79d1\u5b66\uff0c\u5373\u628a\u4ee3\u6570\u63a8\u5e7f\u5230\u62bd\u8c61\u4ee3\u6570[4]\u3002<\/p>\n\n\n\n(4) \u7ebf\u6027\u4ee3\u6570<\/h4>\n\n\n\n
\u7ebf\u6027\u4ee3\u6570\u662f\u62bd\u8c61\u4ee3\u6570\u7279\u6b8a\u7684\u4e00\u7c7b\uff0c\u5176\u4ee3\u6570\u7ed3\u6784\u4e3a\uff1a\u5411\u91cf\u7a7a\u95f4(vector spaces\uff0c\u4e5f\u53eb\u7ebf\u6027\u7a7a\u95f4) + \u7ebf\u6027\u53d8\u6362(linear mappings)\u3002\u5f88\u5bb9\u6613\u5c06\u7ebf\u6027\u4ee3\u6570\u548c\u77e9\u9635\u7406\u8bba\u7b49\u540c\u8d77\u6765\uff0c\u4f46\u5176\u5b9e\u662f\u4e0d\u4e00\u6837\u7684\uff0c\u8ba8\u8bba\u7ebf\u6027\u53d8\u6362\u662f\u57fa\u4e8e\u9009\u5b9a\u4e00\u7ec4\u57fa\u7684\u524d\u63d0\u4e0b\u3002<\/p>\n\n\n\n
\u6458\u6284mathoverflow\u4e0a\u7684\u4e00\u4e2a\u56de\u7b54(\u539f\u6587\u5728\u8fd9\u91cc)\uff1a<\/p>\n\n\n\n
When you talk about matrices, you\u2019re allowed to talk about things like the entry in the 3rd row and 4th column, and so forth. In this setting, matrices are useful for representing things like transition probabilities in a Markov chain, where each entry indicates the probability of transitioning from one state to another.\n\nIn linear algebra, however, you instead talk about linear transformations, which are not a list of numbers, although sometimes it is convenient to use a particular matrix to write down a linear transformation. However, when you\u2019re given a linear transformation, you\u2019re not allowed to ask for things like the entry in its 3rd row and 4th column because questions like these depend on a choice of basis. Instead, you\u2019re only allowed to ask for things that don\u2019t depend on the basis, such as the rank, the trace, the determinant, or the set of eigenvalues. This point of view may seem unnecessarily restrictive, but it is fundamental to a deeper understanding of pure mathematics<\/code><\/pre>\n\n\n\n1.2 \u4ee3\u6570\u7ed3\u6784<\/h3>\n\n\n\n
\u65e2\u7136\u62bd\u8c61\u4ee3\u6570\u7814\u7a76\u5bf9\u8c61\u662f\u4ee3\u6570\u7ed3\u6784(algebraic structure)\uff0c\u90a3\u4ec0\u4e48\u662f\u4ee3\u6570\u7ed3\u6784\u5462\u3002\u770b\u4e86\u591a\u4e2a\u4e0d\u540c\u89d2\u5ea6\u63cf\u8ff0\u4ee3\u6570\u7ed3\u6784\uff0c\u5982\u767e\u5ea6\u767e\u79d1\u4ee3\u6570\uff1a\u4ee3\u6570\u662f\u7814\u7a76\u6570\u3001\u6570\u91cf\u3001\u5173\u7cfb\u4e0e\u7ed3\u6784\u7684\u6570\u5b66\u5206\u652f\u3002\u8fd8\u662f\u89c9\u5f97<\/p>\n\n\n\n
\u300a[\u8f6c]MIT\u725b\u4eba\u89e3\u8bf4\u6570\u5b66\u4f53\u7cfb\u300b\u4e2d\u7684\u63cf\u8ff0\u6700\u6df1\u5165\u6d45\u51fa\uff0c\u5982\u4e0b\uff1a<\/p>\n\n\n\n
\u4ee3\u6570\u4e3b\u8981\u7814\u7a76\u7684\u662f\u8fd0\u7b97\u89c4\u5219\u3002\u4e00\u95e8\u4ee3\u6570\uff0c \u5176\u5b9e\u90fd\u662f\u4ece\u67d0\u79cd\u5177\u4f53\u7684\u8fd0\u7b97\u4f53\u7cfb\u4e2d\u62bd\u8c61\u51fa\u4e00\u4e9b\u57fa\u672c\u89c4\u5219\uff0c\u5efa\u7acb\u4e00\u4e2a\u516c\u7406\u4f53\u7cfb\uff0c\u7136\u540e\u5728\u8fd9\u57fa\u7840\u4e0a\u8fdb\u884c\u7814\u7a76\u3002\u4e00\u4e2a\u96c6\u5408\u518d\u52a0\u4e0a\u4e00\u5957\u8fd0\u7b97\u89c4\u5219\uff0c\u5c31\u6784\u6210\u4e00\u4e2a\u4ee3\u6570\u7ed3\u6784\u3002<\/code><\/pre>\n\n\n\n1.3 \u521d\u7b49\u4ee3\u6570\u2013>\u62bd\u8c61\u4ee3\u6570<\/h3>\n\n\n\n
\u62bd\u8c61\u4ee3\u6570\u5c06\u521d\u7b49\u4ee3\u6570\u7684\u4e00\u4e9b\u6982\u5ff5\u5ef6\u4f38\u3002<\/p>\n\n\n\n
(1) \u6570 \u2013> \u96c6\u5408<\/h4>\n\n\n\n
\u96c6\u5408\u5728\u6734\u7d20\u96c6\u5408\u8bba(naive set theory)\u548c\u516c\u7406\u5316\u96c6\u5408\u8bba(axiomatic set theory)\u7684\u5b9a\u4e49\u662f\u4e0d\u4e00\u6837\u7684\uff0c\u524d\u8005\u6307\u7531\u4e00\u4e9b\u5143\u7d20\u7ec4\u6210\uff1b\u540e\u8005\u6307\u5177\u6709\u67d0\u79cd\u7279\u5b9a\u6027\u8d28\u4e8b\u7269\u7684\u603b\u4f53\u3002\u8fd8\u662f\u770b\u7ef4\u57fa\u82f1\u6587\u8bcd\u6761\u5427[7]\uff1a<\/p>\n\n\n\n
Rather than just considering the different types of numbers, abstract algebra deals with the more general concept of sets: a collection of all objects (called elements) selected by property specific for the set.<\/code><\/pre>\n\n\n\n(2) + \u2013> \u4e8c\u5143\u8fd0\u7b97<\/h4>\n\n\n\n
\u52a0\u53f7+\u88ab\u62bd\u8c61\u4e3a\u4e8c\u5143\u8fd0\u7b97*(binary operation)\uff0c\u5bf9\u4e24\u4e2a\u5143\u7d20\u4f5c\u4e8c\u5143\u8fd0\u7b97\uff0c\u5f97\u5230\u7684\u65b0\u5143\u7d20\u4ecd\u7136\u5c5e\u4e8e\u8be5\u96c6\u5408\uff0c\u8fd9\u53eb\u5c01\u95ed\u6027(closure)<\/strong>\u3002\u5b9e\u9645\u4e0a\uff0c\u52a0\u51cf\u4e58\u9664\u90fd\u53eb\u4e8c\u5143\u8fd0\u7b97(\u4e8c\u5143\u6307\u7684\u662f\u4e24\u4e2a\u64cd\u4f5c\u6570)\u3002
\u539f\u6587\u5982\u4e0b[7]\uff1a<\/p>\n\n\n\nThe notion of addition (+) is abstracted to give a binary operation, \u2217 say. The notion of binary operation is meaningless without the set on which the operation is defined. For two elements a and b in a set S, a \u2217 b is another element in the set; this condition is called closure.\n\nAddition (+), subtraction (-), multiplication (\u00d7), and division (\u00f7) can be binary operations when defined on different sets, as are addition and multiplication of matrices, vectors, and polynomials.<\/code><\/pre>\n\n\n\n(3) 0\/1 \u2013> \u5355\u4f4d\u5143<\/h4>\n\n\n\n
0\u548c1\u88ab\u62bd\u8c61\u6210\u5355\u4f4d\u5143<\/strong>(identity elements)\uff0c0\u4e3a\u52a0\u6cd5\u5355\u4f4d\u5143\uff0c1\u4e3a\u4e58\u6cd5\u5355\u4f4d\u5143\u3002<\/p>\n\n\n\n\u5355\u4f4d\u5143\u662f\u96c6\u5408\u7684\u4e00\u4e2a\u7279\u6b8a\u5143\u7d20(\u8ddf\u4e8c\u5143\u8fd0\u7b97\u6709\u5173)\uff0c\u6ee1\u8db3\u5355\u4f4d\u5143\u4e0e\u5176\u4ed6\u5143\u7d20\u76f8\u7ed3\u5408\u65f6\uff0c\u4e0d\u6539\u53d8\u8be5\u5143\u7d20\uff0c\u5373\u6ee1\u8db3a \u2217 e = a \u4e0e e \u2217 a = a\u3002\u53ef\u89c1\uff0c\u5355\u4f4d\u5143\u53d6\u51b3\u4e8e\u5143\u7d20\u4e0e\u4e8c\u5143\u8fd0\u7b97<\/strong>\uff0c\u5982\u77e9\u9635\u7684\u52a0\u6cd5\u5355\u4f4d\u5143\u662f\u96f6\u77e9\u9635\uff0c\u77e9\u9635\u7684\u4e58\u6cd5\u5355\u4f4d\u5143\u662f\u5355\u4f4d\u77e9\u9635\u3002<\/p>\n\n\n\n\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u6709\u4e9b\u96c6\u5408\u4e0d\u5b58\u5728\u5355\u4f4d\u5143<\/strong>\uff0c\u5982\u6b63\u6574\u6570\u96c6\u5408(the set of positive natural numbers)\u6ca1\u6709\u52a0\u6cd5\u5355\u4f4d\u5143(no identity element for addition)\u3002<\/p>\n\n\n\n\u7ef4\u57fa\u767e\u79d1\u539f\u6587\u5982\u4e0b\uff1a<\/p>\n\n\n\n
The numbers zero and one are abstracted to give the notion of an identity element for an operation. An identity element is a special type of element of a set with respect to a binary operation on that set. For a general binary operator \u2217 the identity element e must satisfy a \u2217 e = a and e \u2217 a = a. Not all sets and operator combinations have an identity element.<\/code><\/pre>\n\n\n\n(4) \u8d1f\u6570 \u2013> \u9006\u5143\u7d20\uff08\u9006\u5143\uff09<\/h4>\n\n\n\n
\u8d1f\u6570\u63a8\u5e7f\u5230\u9006\u5143\u7d20<\/strong>(inverse element)\uff0c\u5bf9\u4e8e\u52a0\u6cd5\uff0ca\u7684\u9006\u5143\u7d20\u662f-a\uff1b\u5bf9\u4e8e\u4e58\u6cd5\uff0ca\u7684\u9006\u5143\u7d20\u662f\u5012\u65701\/a\u3002<\/p>\n\n\n\n\u76f4\u89c2\u5730\u8bf4\uff0c\u9006\u5143\u53ef\u4ee5\u64a4\u9500\u64cd\u4f5c\uff0c\u5982\u52a0\u4e86\u4e00\u4e2a\u6570a\uff0c\u518d\u52a0\u4e0a\u8be5\u6570\u7684\u9006\u5143-a(\u76f8\u5f53\u4e8e\u64a4\u6d88\u64cd\u4f5c)\uff0c\u7ed3\u679c\u8fd8\u662f\u4e00\u6837\u3002<\/p>\n\n\n\n
\u7ef4\u57fa\u767e\u79d1\u539f\u6587\u5982\u4e0b\uff1a<\/p>\n\n\n\n
The negative numbers give rise to the concept of inverse elements. For addition, the inverse of a is written \u2212a, and for multiplication the inverse is written a\u22121. A general two-sided inverse element (left inverse + right inverse) a\u22121 satisfies the property that a \u2217 a\u22121 = 1 and a\u22121 \u2217 a = 1.\n\nThe idea of an inverse element generalises concepts of a negation (sign reversal) in relation to addition, and a reciprocal in relation to multiplication. The intuition is of an element that can \u2018undo\u2019 the effect of combination with another given element. While the precise definition of an inverse element varies depending on the algebraic structure involved, these definitions coincide in a group.<\/code><\/pre>\n\n\n\n(5) \u7ed3\u5408\u5f8b<\/h4>\n\n\n\n
\u7ed3\u5408\u5f8b(Associative property)\u662f\u67d0\u4e9b\u4e8c\u5143\u8fd0\u7b97\u7684\u6027\u8d28\uff0c\u6709\u4e9b\u4e8c\u5143\u8fd0\u7b97\u6ca1\u6709\u7ed3\u5408\u5f8b(\u5982\u51cf\u6cd5\u3001\u9664\u6cd5\u3001\u516b\u5143\u6570)\u3002\u539f\u6587\u5982\u4e0b\uff1a<\/p>\n\n\n\n
Addition of integers has a property called associativity. That is, the grouping of the numbers to be added does not affect the sum. In general, this becomes (a \u2217 b) \u2217 c = a \u2217 (b \u2217 c)<\/strong>. This property is shared by most binary operations, but not subtraction or division or octonion multiplication.<\/code><\/pre>\n\n\n\n(6) \u4ea4\u6362\u5f8b<\/h4>\n\n\n\n
\u4ea4\u6362\u5f8b(Commutative property)\uff0c\u6539\u53d8\u4e8c\u5143\u8fd0\u7b97\u7b26\u4e24\u8fb9\u7684\u5143\u7d20\u4e0d\u5f71\u54cd\u7ed3\u679c\u3002\u5e76\u4e0d\u662f\u6240\u6709\u4e8c\u6b21\u5143\u8fd0\u7b97\u90fd\u6ee1\u8db3\u4ea4\u6362\u5f8b(\u5982\u77e9\u9635\u7684\u4e58\u6cd5)\u3002<\/p>\n\n\n\n
\u7ef4\u57fa\u767e\u79d1\u539f\u6587\u5982\u4e0b\uff1a<\/p>\n\n\n\n
A binary operation is commutative if changing the order of the operands does not change the result. Addition and multiplication of real numbers are both commutative. In general, this becomes a \u2217 b = b \u2217 a<\/strong>\n\nThis property does not hold for all binary operations. For example, matrix multiplication and quaternion multiplication are both non-commutative.\n<\/code><\/pre>\n\n\n\n2. Group-like<\/h2>\n\n\n\n
\u4ee3\u6570\u7ed3\u6784(R, *)\uff0c\u4e8c\u5143\u8fd0\u7b97\u6839\u636e\u5c01\u95ed\u6027\u3001\u5355\u4f4d\u5143\u3001\u9006\u5143\u3001\u7ed3\u5408\u5f8b\u3001\u4ea4\u6362\u5f8b\uff0c\u53ef\u4ee5\u5f52\u7eb3\u6210\u4e0d\u540c\u7684\u7fa4\u3002<\/p>\n\n\n\n
\u672c\u8282\u4ecb\u7ecd\u7684group-like\uff0c\u4ece\u6700\u4e0d\u4e25\u683c\u5230\u4e25\u683c(\u4f9d\u6b21\u6dfb\u52a0\u9650\u5236\u6761\u4ef6)\uff0c\u5176\u5173\u7cfb\u56fe\u5982\u4e0b\uff1a<\/p>\n\n\n\n
![\"\"\/](\"http:\/\/sparkandshine.net\/wordpress\/wp-content\/uploads\/2015\/02\/image_thumb14.png\")
\u56fe1 \u7fa4\u4e4b\u95f4\u7684\u5173\u7cfb(\u8d85\u7ea7\u597d\u7684\u56fe<\/figcaption><\/figure><\/div>\n\n\n\n\u7ef4\u57fa\u767e\u79d1\u6709\u4e00\u5f20\u8868\uff0c\u7ed9\u51fa\u66f4\u8be6\u7ec6\u7684group-like\u95f4\u7684\u5173\u7cfb\uff0c\u5982\u4e0b\uff1a<\/p>\n\n\n\n![\"\"\/](\"http:\/\/sparkandshine.net\/wordpress\/wp-content\/uploads\/2015\/02\/image_thumb15.png\")
\u56fe2 Group-like structures (source from here)<\/figcaption><\/figure>\n\n\n\n2.1 \u539f\u7fa4<\/h3>\n\n\n\n
\u539f\u7fa4(magma)<\/strong>\u662f\u4e00\u79cd\u57fa\u672c\u7684\u4ee3\u6570\u7ed3\u6784\uff0c\u53ea\u8981\u6ee1\u8db3\u4e24\u5143\u7d20\u4f5c\u4e8c\u5143\u8fd0\u7b97\u5f97\u5230\u65b0\u5143\u7d20\u4ecd\u5c5e\u4e8e\u8be5\u96c6\u5408\uff0c\u5373\u5c01\u95ed\u6027<\/strong>\u3002<\/p>\n\n\n\n\u7ef4\u57fa\u767e\u79d1\u539f\u6587\u5982\u4e0b\uff1a<\/p>\n\n\n\n
A magma is a basic kind of algebraic structure. Specifically, a magma consists of a setMequipped with a single binary operationM \\times M \\rightarrow M. The binary operation must be closed by definition but no other properties are imposed.<\/p>\n\n\n\n
2.2 \u534a\u7fa4<\/h3>\n\n\n\n
\u534a\u7fa4(Semigroup)<\/strong>\uff0c\u6ee1\u8db3\u7ed3\u5408\u5f8b(associative property)\u7684\u4ee3\u6570\u7ed3\u6784\u3002V=<S\uff0c* >\uff0c\u5176\u4e2d\u4e8c\u5143\u8fd0\u7b97\u662f\u53ef\u7ed3\u5408\u7684\uff0c\u5373(ab)c=a(b*c)<\/em>\uff0c\u5219\u79f0V\u662f\u534a\u7fa4<\/strong>\u3002<\/p>\n\n\n\n\u7ef4\u57fa\u767e\u79d1\u539f\u6587\u5982\u4e0b\uff1a<\/p>\n\n\n\n
A semigroup is an algebraic structure consisting of a set together with an associative binary operation.<\/p>\n\n\n\n
A semigroup generalizes a monoid in that a semigroup need not have an identity element. It also (originally) generalized a group (a monoid with all inverses) in that no element had to have an inverse, thus the name semigroup.<\/p>\n\n\n\n
2.3 \u5e7a\u534a\u7fa4<\/h3>\n\n\n\n
\u5e7a\u534a\u7fa4(monoid)<\/strong>\u5728\u534a\u7fa4\u7684\u57fa\u7840\u4e0a\uff0c\u8fd8\u9700\u8981\u6ee1\u8db3\u6709\u4e00\u4e2a\u5355\u4f4d\u5143<\/strong>\u3002<\/p>\n\n\n\n\u7ef4\u57fa\u767e\u79d1\u539f\u6587\u5982\u4e0b\uff1a<\/p>\n\n\n\n
A monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are semigroups with identity.<\/p>\n\n\n\n
2.4 \u7fa4<\/h3>\n\n\n\n
\u7fa4(group)\u662f\u4e24\u4e2a\u5143\u7d20\u4f5c\u4e8c\u5143\u8fd0\u7b97\u5f97\u5230\u7684\u4e00\u4e2a\u65b0\u5143\u7d20\uff0c\u9700\u8981\u6ee1\u8db3\u7fa4\u516c\u7406<\/strong>(group axioms)\uff0c\u5373\uff1a<\/p>\n\n\n\n- \u5c01\u95ed\u6027\uff1aa \u2217 b is another element in the set<\/li>
- \u7ed3\u5408\u5f8b\uff1a(a \u2217 b) \u2217 c = a \u2217 (b \u2217 c)<\/li>
- \u5355\u4f4d\u5143\uff1aa \u2217 e = a and e \u2217 a = a<\/li>
- \u9006 \u5143\uff1a\u52a0\u6cd5\u7684\u9006\u5143\u4e3a-a\uff0c\u4e58\u6cd5\u7684\u9006\u5143\u4e3a\u5012\u65701\/a\uff0c\u2026 (\u5bf9\u4e8e\u6240\u6709\u5143\u7d20)
<\/li><\/ul>\n\n\n\n\u5982\u6574\u6570\u96c6\u5408\uff0c\u4e8c\u6b21\u5143\u8fd0\u7b97\u4e3a\u52a0\u6cd5\u5c31\u662f\u4e00\u4e2a\u7fa4(\u5c01\u95ed\u6027\u662f\u663e\u7136\u7684\uff0c\u52a0\u6cd5\u6ee1\u8db3\u7ed3\u5408\u5f8b\uff0c\u5355\u4f4d\u5143\u4e3a0\uff0c\u9006\u5143\u53d6\u76f8\u53cd\u6570-a)\u3002
\u7ef4\u57fa\u767e\u79d1\u539f\u6587\u5982\u4e0b\uff1a<\/p>\n\n\n\n
A group is a set of elements together with an operation that combines any two of its elements to form a third element satisfying four conditions called the group axioms, namely closure, associativity, identity and invertibility.<\/p>\n\n\n\n
One of the most familiar examples of a group is the set of integers together with the addition operation<\/p>\n\n\n\n
2.5 \u963f\u8d1d\u5c14\u7fa4(\u4ea4\u6362\u7fa4)<\/h4>\n\n\n\n
\u963f\u8d1d\u5c14\u7fa4(Abelian Group)\u5728\u7fa4\u7684\u57fa\u7840\u4e0a\uff0c\u8fd8\u9700\u6ee1\u8db3\u4ea4\u6362\u5f8b<\/strong>\u3002\u5982\u6574\u6570\u96c6\u5408\u548c\u52a0\u6cd5\u8fd0\u7b97\uff0c(Z,+)\uff0c\u662f\u4e00\u4e2a\u963f\u8d1d\u5c14\u7fa4\u3002<\/p>\n\n\n\n\u7fa4\u516c\u7406\uff1a\u89c12.4 \u7fa4\u3002
\u4ea4\u6362\u5f8b\uff1aa + b = b + a
\u7ef4\u57fa\u767e\u79d1\u539f\u6587\u5982\u4e0b\uff1a<\/p>\n\n\n\n
An abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order (the axiom of commutativity).<\/p>\n\n\n\n
3. \u73af\u8bba<\/h2>\n\n\n\n
\u73af\u5728\u4ea4\u6362\u7fa4\u57fa\u7840\u4e0a\uff0c\u8fdb\u4e00\u6b65\u9650\u5236\u6761\u4ef6\u3002\u73af\u3001\u4ea4\u6362\u73af\u3001\u57df\u95f4\u7684\u5173\u7cfb\u5982\u4e0b\uff1a<\/p>\n\n\n\n![\"\"\/](\"http:\/\/sparkandshine.net\/wordpress\/wp-content\/uploads\/2015\/02\/image_thumb16.png\")
\u56fe3 \u73af\u3001\u4ea4\u6362\u73af\u3001\u57df\u95f4\u7684\u5173\u7cfb<\/figcaption><\/figure>\n\n\n\n3.1 \u73af<\/h3>\n\n\n\n
\u73af(ring)\u5728\u963f\u8d1d\u5c14\u7fa4(\u4e5f\u53eb\u4ea4\u6362\u7fa4)\u7684\u57fa\u7840\u4e0a\uff0c\u6dfb\u52a0\u4e00\u79cd\u4e8c\u5143\u8fd0\u7b97\u00b7(\u867d\u53eb\u4e58\u6cd5\uff0c\u4f46\u4e0d\u540c\u4e8e\u521d\u7b49\u4ee3\u6570\u7684\u4e58\u6cd5<\/strong>)\u3002<\/p>\n\n\n\n\u4e00\u4e2a\u4ee3\u6570\u7ed3\u6784\u662f\u73af(R, +, \u00b7)\uff0c\u9700\u8981\u6ee1\u8db3\u73af\u516c\u7406(ring axioms)<\/strong>\uff0c\u5982(Z,+, \u22c5)\u3002<\/p>\n\n\n\n\u73af\u516c\u7406\u5982\u4e0b\uff1a<\/p>\n\n\n\n
(1) (R, +)\u662f\u4ea4\u6362\u7fa4<\/p>\n\n\n\n
![\"\"](\"http:\/\/43.142.23.155\/wp-content\/uploads\/2022\/03\/image-18.png\")