basis/injection vs canonical basis/injection

Question

Let $\mathbb R^n$ (or $\mathbb C^n$) be a finite dimensional vector space over $\mathbb R$ (or $\mathbb C$). Some texts in linear algebra used that there is a canonical basis of $V$. (Unfortunately, without mentioning specific definitions..)

My Questions:

(1) What is a difference between canonical basis and just basis? (Or what is a canonical basis? Is every basis is a canonical basis?)

(2) What is a canonical injection? (My understanding is if $f:X\to Y$ is one-one function, we may say that $f$ is injective or $f$ is injection. Please correct me if I am wrong! Is every injection is a canonical injection? I am wondering why do we need the word canonical? Any motivation?)

Just as an example: The book "Algebras of Linear Transformation" by Farenick, Douglas R. uses the terms like canonical basis and canonical injections.

Answer 1

I don't have access to that book, but I find it very hard to believe that it says that every finite-dimensional vector space over $\mathbb R$ or $\mathbb C$ has a canonical basis. The space $\mathbb{K}^n$ (where $\mathbb K$ is $\mathbb R$ or $\mathbb C$) has a canonical basis:$$e_1=(1,0,0,\ldots,0),e_2=(0,1,0,\ldots,0),\ldots$$Also, the space $\mathbb{K}_n[x]$ of all polynomials in the variable $x$ whose degree is smaller thatn or equal to $n$ has a canonical basis: $1,x,x^2,\ldots,x^n$. But there is no general concept of canonical basis.

I am also not aware of a general concept of canonical injection. You must be missing something here.

Answer 2

Let's write vectors in $F^n$, where $F$ is a field, as rows (just for ease of typesetting them).

If $V$ is an $F$-vector space and $B=(v_1,v_2,\dots,v_n)$ is a basis (together with an ordering, so we distinguish between bases where even just the order differ). We can define a map $$ C_B\colon V\to F^n $$ that maps each vector $v\in V$ to the (unique) vector $(\alpha_1,\dots,\alpha_n)\in F^n$ such that $$ v=\alpha_1v_1+\alpha_2v_2+\dots+\alpha_nv_n $$ This is the coordinate map relative to the basis $B$ and $C_B(v)$ is the coordinate vector of $v$ (terminology is not universal). In other words $$ C_B(v)=(\alpha_1,\dots,\alpha_n) \quad\text{if and only if}\quad v=\alpha_1v_1+\alpha_2v_2+\dots+\alpha_nv_n $$ Such a map is a linear bijection.

Prove that if $B$ and $B'$ are different bases of $V$, then the coordinate maps $C_B$ and $C_{B'}$ differ.

If $V=F^n$ we might ask whether there exists a basis $E=(e_1,\dots,e_n)$ such that $C_E$ is the identity map. If such a basis exists, it is unique, and, indeed, it exists. Take $$ e_1=(1,0,0,\dots,0),\quad e_2=(0,1,0,\dots,0),\quad \dots,\quad e_n=(0,0,\dots,0,1) $$ and consider that, when $B$ is a basis of $V$ as before, then $$ C_B(v_1)=e_1,\quad C_B(v_2)=e_2,\quad \dots,\quad C_B(v_n)=e_n $$ so now the distinguished property of $E$ is evident.

This is why such a basis is called canonical: the coordinate vector of $v\in F^n$ relative to $E$ is the vector $v$ itself.

Such a concept is meaningless if $V$ is not $F^n$, because in this case $v$ and $C_B(v)$ live in different spaces.

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The term canonical injection refers to a slightly different framework: if $U$ is a subspace of $V$, then the canonical injection is the (linear) map that sends each vector in $U$ to itself (as an element of $V$). In general there are several injective linear maps $U\to V$, and just one is “distinguished” and is thus called canonical.

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Personal note. I beg to disagree with the other answer in the part where it claims that the space of polynomials of degree up to $n-1$ has a canonical basis. The basis $(1,x,x^2,\dots,x^{n-1})$ is an obvious choice, but it cannot be selected among other bases just by vector space properties, contrary to the canonical basis of $F^n$, at least not by “elementary” properties.