Linear algebra notes (vector)

1) Quantity: Also called scalar, pure quantity, only the size has no direction, can use a numerical value to determine;
2) Vector: Also known as vector, describing this kind of quantity not only needs the size, but also needs to express its direction;
3) directed segment: a segment with a certain length and a definite direction;
4) geometric vector: short vector, Vector called geometric vector represented by a directed segment;
5) fixed vector, free vector: whether the starting point is fixed to differentiate, the starting point fixed is called the fixed vector, the beginning is not fixed called the free vector;
6) Vector equal: the size and direction are the same;
7) Negative vectors: The same size, the opposite direction;
8) Modulo: Also called norm, for the size of the vector; a| |
9) unit vector: vector of modulo 1;
10) 0 Vector: the size (length) is 0 vector, its starting point and end point coincident, direction is uncertain;
11) Addition of vectors: parallelogram principle;
12) Subtraction of vectors: the inverse operation of addition, the same applies to parallelogram principle;
13) Multiplication: The number of multiplication, the direction is constant or reverse, the size is k times;
14) Linear operation properties: A+b=b+a; (a+b) +c=a+ (B+C); A+0=a; A + (-a) = 0; K (a+b) =ka+kb; (k+l) A=ka+la; (KL) a=k (LA); 1?a=a
15) vector space: Also known as geometric vector space. If the vector set V is closed to addition and multiplication operations, then v is called vector space;
16) Two-dimensional vector space, three-dimensional vector space
17) coordinate notation for geometric vectors: α=aex+bey= (A, b). | α| | ^2=a^2+b^2;
18) Coordinate vector addition: the sum of the vectors is equal to the component, and the vector number multiplication equals the component number multiplication;
n element Array vector: Also called n-ary vector, consisting of n ordered number (is arranged, not combination);
N-ary vector: 0 vectors, vector addition equals component addition, vector number multiplication equals component number multiplication, with eight basic operation properties listed above;
21) vector space: A more general definition, if V is an n-ary vector set, the component of its vector is the number of Number field K, if V is closed to addition, the K number in the log domain is closed, then v is the vector space. Also known as the n-dimensional vector space on a number of fields K;
22) Linear combination: K1...KS is the number of fields K, α1 ... Αs is the n vector space on a number of fields K, then K1?α1+k2?α2+...+ks?αs is α1. A linear combination of AS., linear: Α=k1?α1+k2?α2+...+ks?αs,α belongs to V;
23) Linear correlation: If for vector group α1 ... Αs, the existence of K1...KS is not completely zero so that k1?α1+k2?α2+...+ks?αs=0 established, it is called α1. The αs is linearly correlated, otherwise linearly independent;
24) linear correlation and linear independent judgment can be judged by the solution of the singular linear equations: only 0 solutions are linearly independent, otherwise they are linearly correlated.
25) The equations of the vector group are expressed as vector columns, the number of vectors equals the number of unknown quantities, and the number of vector components is the number of equations;
26) Single 0 vector linear correlation, single non-zero vector linearly independent;
27) The sufficient and necessary condition for the linear correlation of a vector group is that one of the vectors can be linearly represented by the remaining vectors;
28) The sufficient and necessary condition for the linear correlation of two vectors is that their components correspond proportionally;
29) The vector group is linearly correlated with the partial vector, then the whole vector group is linearly correlated;
30) maximal linear independent group, rank of vector Group;
31) The vector group is equivalent: the mutual linear table out, the vector group equivalence has the transitivity, the equivalent linear independent group contains the same number of vectors; all the maximal linear independent groups of a vector group contain the same number of vectors;
32) The rank of the vector group: the number of vectors contained in the maximal linear independent group of the vector group; if Vector Group A can have a linear table of vector Group B, then the rank of a is less than or equal to the rank of B;
33) Two vector groups are equivalent, then the rank is the same; the sufficient and necessary condition for the linear independence of a vector group is that its rank equals the number of vectors it contains;
Vector algorithms are relatively simple, and the algorithm for finding maximal linear independent groups including the rank of vectors can be found in mymathlib.

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Linear algebra notes (vector)