How should I read/interpret this notation?

Question

The notation used is $\boldsymbol{v} \mapsto [\boldsymbol{v}]_B$, I've read it before but I don't really understand. It appears in the following proof for context:

Two finite-dimensional vector spaces $V$ and $W$ with the same dimension are isomorphic.

Proof: Assume $\dim V=\dim W=n$. Let $\boldsymbol{b_1},\ldots,\boldsymbol {b_n}$ be a base for $V$. Then $\boldsymbol{v} \mapsto[\boldsymbol{v}]_B$ is an isomorphism between $V$ and $\mathbb{R}^n$, etc...

My guess is it's the mapping of the vector $\boldsymbol{v}$ as a linear combination of the basis vectors in $\boldsymbol{B}$?

Answer 1

The vector $v$ lies in the vector space $V$ of dimension $n$. Once you put a coordinate system on $V$ (which is to say, once you choose a base $B$), then you can find the coordinates of the vector $v$. This is what they mean with $[v]_B$. So we have

• $v$: A vector in the vector space $V$
• $[v]_B$: The coordinates of the vector $v$ using the base $B$, which is to say the tuple $(v_1, v_2, \ldots,v_n)$ of numbers such that $$v = v_1b_1+v_2b_2 + \cdots v_nb_n$$

Using the base $B$ it is not difficult to translate between these two. That's what $v\mapsto [v]_B$ means.

Answer 2

Although I do not know this notation it is clear from the context that $ v\to[ v]_B$ denotes the coordinate isomorphism, so if $B=(b_1,\ldots,b_n)$ is a basis of $V$, then $v$ has a unique linear combination $v=a_1b_1+\ldots+a_nb_n$, where $a_i$ are scalars from your base field $K$ ($\mathbb R$ in your case). Now $[v]_B = (a_1,\ldots,a_n)\in K^n$.

Easier to define is the inverse: $$ K^n\ni (a_1,\ldots,a_n) \to a_1b_1+\ldots+a_nb_n\in V$$