Linear Algebra basics: vector combination

1. Linear Combination

For n-dimensional vector groups {V1, V2, V3 ,..., VK}, if any real number C1, C2, C3 ,... CK, then the vector V' = C1 * V1 + C2 * V2 +... + ck * VK is a linear combination of vector groups. a linear combination refers to a combination of addition and multiplication operations.

2. Vector Space
A vector set composed of all Linear Combinations in a vector group. It is called a vector space formed by a vector group. Vector V1 and V2 can be regarded as span (V1, V2 ).
If V1 is a two-dimensional vector, span (V1) = {cv1} is a straight line.
V2 is also a two-dimensional vector, so span (V1, V2) = R2, provided that the direction of V1 and V2 is inconsistent.
Span (V1, V2) = R2, because any vector in R2 can be expressed as C1 * V1 + C2 * V2, but the method in it is not intuitive. Usually expressed by two orthogonal unit vectors () and.

3. Linear Independence
Vector V1, V2 ,..., VK. If any vector cannot be expressed as a linear combination of other vectors, it is called linear correlation.
From span (V1, V2 ,..., from the VK perspective, if these vectors are not linear-independent, there must be redundant vectors. after deducting these redundant vectors, the span will not change.

An equivalent definition of linear independence is that for vector groups V1, V2 ,..., VK, no real numbers C1, C2 ,..., CK (CI is not all 0) causes C1 * V1 + C2 * V2 + ,..., + ck * VK = 0. If a vector is expanded, a K-element equations can be obtained. This definition is converted to "the equations do not have a non-zero solution ", in this way, we can calculate whether a vector group is linearly independent by solving the equation.

4. Linear subspaces of Rn
RN is a set of all n-dimensional vectors, so the sub-space is a subset of RN.
If V is a subset of Rn, and for any V1, V2 ,..., the linear combination of VK is still a member of V (that is, the logarithm multiplication of V, and the addition operation is closed), and V is called the subspace of RN. It is also called a Linear Subspace.
The space produced by arbitrary vectors must be the space of RN.

5. Basis of sub-Spaces
For the span (V1, V2 ,..., VN), If V1, V2 ,..., vn is linearly independent, so V1, V2 ,..., vn is the basis of the sub-space. In other words, the basis of the sub-space is a set of smallest vectors that can be split into sub-spaces.
Any vector in the sub-space can uniquely represent a linear combination of vectors.
There are more than one group of basis in the sub-space. In fact, there are no arrays.