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本文为《Linear algebra and its applications》的读书笔记

Coordinate systems

• An important reason for specifying a basis $\mathcal{B}$ for a vector space $V$ is to impose a “coordinate system” on $V$ .
• This section will show that if $\mathcal{B}$ contains $n$ vectors, then the coordinate system will make $V$ act like ${\mathbb{R}}^n$. If $V$ is already ${\mathbb{R}}^n$ itself, then $\mathcal{B}$ will determine a coordinate system that gives a new “view” of $V$ .

• The existence of coordinate systems rests on the following fundamental result.

• Since $\mathcal{B}$ spans $V$, there exist scalars such that (1) holds. Suppose $\mathbf{x}$ also has the representation
  Then, subtracting, we have
  Since $\mathcal{B}$ is linearly independent, the weights must all be zero. That is, $c_j=d_j$ for $1\le j\le n$.

Be careful not to use a matrix in the proof. The vectors ${\mathbf{v}}_1,...,{\mathbf{v}}_n$ cannot be arranged as the columns of an ordinary matrix when the vectors are in some abstract vector space.

• If $c_1,...,c_n$ are the $\mathcal{B}$-coordinates of $\mathbf{x}$, then the vector in ${\mathbb{R}}^n$
  is the coordinate vector of $\mathbf{x}$ (relative to $\mathcal{B}$, or the $\mathcal{B}$-coordinate vector of $\mathbf{x}$). The mapping $\mathbf{x}\mapsto [\mathbf{x}{]}_{\mathcal{B}}$ is the coordinate mapping (determined by $\mathcal{B}$).

A Graphical Interpretation of Coordinates

• A coordinate system on a set consists of a one-to-one mapping of the points in the set into ${\mathbb{R}}^n$.
  • For example, ordinary graph paper provides a coordinate system for the plane when one selects perpendicular axes and a unit of measurement on each axis. Figure 1 shows the vector $\mathbf{x}=\left[\begin{array}{c}1\\ 6\end{array}\right]$. The coordinates 1 and 6 give the location of $\mathbf{x}$ relative to the standard basis { e 1 , e 2 } \{\boldsymbol e_1, \boldsymbol e_2\} {
    e1​,e2​}: 1 unit in the ${\mathbf{e}}_1$ direction and 6 units in the e2 direction.
    
  • Figure 2 shows the vector $\mathbf{x}$ from Figure 1. However, the standard coordinate grid was erased and replaced by a grid especially adapted to the basis B = { [ 1 0 ] , [ 1 2 ] } \mathcal B=\{\begin{bmatrix}1\\0\end{bmatrix},\begin{bmatrix}1\\2\end{bmatrix}\} B={
    [10​],[12​]}. The coordinate vector $[\mathbf{x}{]}_{\mathcal{B}}=\left[\begin{array}{c}-2\\ 3\end{array}\right]$ gives the location of $\mathbf{x}$ on this new coordinate system: -2 units in the ${\mathbf{b}}_1$ direction and 3 units in the ${\mathbf{b}}_2$ direction.
    

Coordinates in ${\mathbb{R}}^n$

• When a basis $\mathcal{B}$ for ${\mathbb{R}}^n$ is fixed, the $\mathcal{B}$-coordinate vector of a specified $\mathbf{x}$ is easily found.

• Let
  Then the vector equation
  is equivalent to
  
• We call $P_B$ the change-of-coordinates matrix (坐标变换矩阵) from $\mathcal{B}$ to the standard basis in ${\mathbb{R}}^n$.
  • Left-multiplication by $P_B$ transforms the coordinate vector $[\mathbf{x}{]}_{\mathcal{B}}$ into $\mathbf{x}$.
  • Since the columns of $P_B$ form a basis for ${\mathbb{R}}^n$, $P_B$ is invertible. Left-multiplication by $P_B^{-1}$ converts $\mathbf{x}$ into its $\mathcal{B}$-coordinate vector:
    The correspondence $\mathbf{x}\mapsto [\mathbf{x}{]}_{\mathcal{B}}$, produced here by $P_B^{-1}$, is the coordinate mapping mentioned earlier. Since $P_B^{-1}$ is an invertible matrix, the coordinate mapping is a one-to-one linear transformation from ${\mathbb{R}}^n$ onto ${\mathbb{R}}^n$. This property of the coordinate mapping is also true in a general vector space that has a basis.

The Coordinate Mapping

• Choosing a basis B = { b 1 , . . . , b n } \mathcal B = \{\boldsymbol b_1,..., \boldsymbol b_n\} B={
  b1​,...,bn​} for a vector space $V$ introduces a coordinate system in $V$ . The coordinate mapping $\mathbf{x}\mapsto [\mathbf{x}{]}_{\mathcal{B}}$ connects the possibly unfamiliar space $V$ to the familiar space ${\mathbb{R}}^n$. See Figure 5.
  

• Take two typical vectors in $V$ , say,
  Then, using vector operations,
  It follows that
  If $r$ is any scalar, then
  So
  Thus the coordinate mapping is a linear transformation. It is easy to show that the coordinate mapping is one-to-one and maps $V$ onto ${\mathbb{R}}^n$.

• The linearity of the coordinate mapping extends to linear combination:
  The coordinate mapping in Theorem 8 is an important example of an isomorphism (同构) from $V$ onto ${\mathbb{R}}^n$.
• In particular, any real vector space with a basis of $n$ vectors is indistinguishable from ${\mathbb{R}}^n$.
  

EXAMPLE 5

• Let $\mathcal{B}$ be the standard basis of the space ${\mathbb{P}}^3$ of polynomials; that is, let B = { 1 , t , t 2 , t 3 } \mathcal B =\{1, t, t^2, t^3\} B={
  1,t,t2,t3}. A typical element $\mathbf{p}$ of ${\mathbb{P}}^3$ has the form
  Thus the coordinate mapping $\mathbf{p}\mapsto [\mathbf{p}{]}_{\mathcal{B}}$ is an isomorphism from ${\mathbb{P}}^3$ onto ${\mathbb{R}}^4$. All vector space operations in ${\mathbb{P}}^3$ correspond to operations in ${\mathbb{R}}^4$.

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