Linear space, subspace, basis, basis coordinates, transition matrix

Definition of Linear Space

It satisfies the addition and multiplication closures. That is to say, all vectors in this space are still in this space after being multiplied by a constant or added to other vectors. It can be further understood that all vectors in this space satisfy the combined closure of addition and multiplication . That is, if V is a linear space, it must first satisfy:

Note: The elements in the linear space are called vectors

Proof of linear space

To prove that V is a linear space on the number field P (expressed as V(P)), it must be verified that V is closed to vector addition and multiplication operations, and satisfies 8 properties;
To explain that V is not a linear space on the number field P, it only needs to explain that V is not closed to one of the addition and multiplication operations of vectors, or the operation does not satisfy one of the eight conditions.

example:

Proof: Theorem 1.1 The linear space V has the only zero element, and any element also has the only negative element.
Note: The zero elements are not necessarily all 0.

common linear space

For example $R^{^{2}}$ , it is a linear space, and the graphic representation is a plane Cartesian coordinate system. Take any vector $\begin{bmatrix} 0\\ 1 \end{bmatrix}$ and $\begin{bmatrix} 1\\ 0 \end{bmatrix}$ do linear combination, $k_{1}$ $\begin{bmatrix} 0\\ 1 \end{bmatrix}$ + $k_{2}$ $\begin{bmatrix} 1\\ 0 \end{bmatrix}$ = $\begin{bmatrix} k_{1}\\ k_{2} \end{bmatrix}$ $\displaystyle \epsilon R^{^{2}}$

{ 0 } (vector 0 ) is also a linear space, and it is the simplest linear space. It is easy to verify that 0 satisfies the closure of addition, multiplication and 8 operation rules. Although it is easy to list two linear spaces, not all All spaces are linear.

nonlinear space

linear subspace

Definition: Let $V_{1}$ be a non-empty subset of the linear space V on the number field K , and satisfy the existing linear operations:
(1) If $x,y\epsilon V_{1}$ , then $(x+y) \epsilon V_{1}$ .
(2) If $x \epsilon V_{1},k \epsilon K$ , then $kx \epsilon V_{1}$ .
Note: (1) and (2) represent the closed principle of addition and multiplication.
is $V_{1}$ called a linear subspace or subspace of V.
If $V_{1}=V or \phi$ ( $\phi$ represents the empty set), $V_{1}$ it is called a trivial subspace; otherwise it is called a non-trivial subspace.