Coordinates translate vectors into scalar lists relative to an ordered basis. For $B=(b_1,\ldots,b_n)$, the coordinate map sends $\sum_ic_ib_i$ to $(c_1,\ldots,c_n)^T$. It is a linear isomorphism with $\mathbb F^n$. Changing basis changes coordinates while leaving the underlying vector unchanged.
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- Definition of coordinates relative to a given basisGiven an ordered basis $B=(b 1,\ldots,b n)$, the coordinate vector $[v] B=(c 1,\ldots,c n)^T$ is defined by $v=\sum...
- Basis exchange using a nonzero coordinateLet $B=(b 1,\ldots,b n)$ and $c=\sum i\gamma ib i$. Replacing $b j$ by $c$ yields a basis iff $\gamma j\ne0$. If it...
- Definition of coordinate vector/mapping/representation relative to a given basisFor basis $B=(b 1,\ldots,b n)$, define $[v] B=(c 1,\ldots,c n)^T$ by $v=\sum ic ib i$. The coordinate mapping $\phi...
- The coordinate vector relative to a given basis is a linear mapping to R^n (or C^n).For a fixed ordered basis, $[au+bv] B=a[u] B+b[v] B$. Expand $u,v$ in the basis and collect coefficients; uniqueness...
- The coordinate vector relative to a given basis is an injective linear mapping to R^n (or C^n).The coordinate mapping is injective: if $[u] B=[v] B$, their unique basis expansions have the same coefficients, so...
- The coordinate vector relative to a given basis is a surjective linear mapping to R^n (or C^n).The coordinate mapping onto $\mathbb F^n$ is surjective: for any tuple $(c 1,\ldots,c n)^T$, the vector $v=\sum ic ib...
- Definition of change of coordinates matrix between two basesFor ordered bases $B=(b j)$ and $C$ of the same space, $P {C\leftarrow B}$ is the matrix whose column $j$ is $[b j]...
- The change of coordinates matrix between two bases exists and is uniqueA change-of-coordinates matrix exists because the coordinate map from $B$ to $C$ is linear. It is unique because its...
- Multiplication by a change of coordinates matrix converts representations for different bases.For ordered bases $B,C$, $[v] C=P {C\leftarrow B}[v] B$. For $B=((1,0),(1,1))$ and standard basis $E$, $P...
- Change of coordinates matrices are invertibleA change-of-coordinates matrix is invertible because coordinate conversion is bijective. Its inverse changes...
- Conjugating by a change of coordinates matrix converts matrix representations with respect to different bases.Let $A B=[T] B$, $A C=[T] C$, and $P=P {B\leftarrow C}$ for an operator $T:V\to V$. Then $A C=P^{-1}A BP$. Thus...
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