Good Notation that clearly distinguish between a vector and its components, a linear transform and its matrix

Question

When we are doing calculations with vectors/matrices, we often need the following things:

• A finite dimensional inner product space $V$ with inner product $\langle\cdot,\cdot\rangle.$
• A basis $\{\mathbf e_j\}$
• A matrix $M$
• A linear transform $T$ that matrix $M$ represents in this given basis.
• A vector $v\in V.$
• Its components $v_j$ in the basis.
• The quadratic form value $\langle v,Tv\rangle.$
• If the basis is orthonormal, we can also write $\langle v,Tv\rangle=\sum M_{ij} v_i v_j.$

So far, everything is clear. There are no ambiguities. However, sometimes, we just say $V=\mathbb R^n,$ and this is where confusion in notation starts:

• First of all, $T$ now has a matrix $A$ in the standard basis, so $\langle v,Tv\rangle= v^T A v.$ Note that now, $v^T$ makes sense as a row vector.
• If we represent the components of $v$ in basis $\{\mathbf e_j\}$ by another column vector $v',$ then we write $v'^T M v'=\langle v,Tv\rangle.$

Now, there are several caveat of such notations:

• $v,v'$ represents the same vector in $\mathbb R^n$, so one feels reluctant to use two different symbols for them.
• The essential issue is, by letting $V=\mathbb R^n,$ we have lost the notation to distinguish between components of a vector in some basis and the vector itself which is independent of the choice of basis - both the vector itself and the list of components in some basis are written as "column vectors". If I just write $$ \begin{bmatrix} 1\\3 \end{bmatrix} $$ then you don't know if this is meant to be a vector in the space $V$ or just the component of some vector in some basis. Every time I write down such a vector I have to say what I mean again and again.
• Of course, the above issue can be resolved by defining explicitly an isomorphism $\varphi: \mathbb R^n \to \mathbb R^n,$ given by $v \mapsto v'$. But this does not seem to be necessary every time, and it further complicates the notation.
• Index notation like $\sum M_{ij} v_i v_j$ is not always possible to use, for example when $v_j$ are given as specific numbers like $\begin{bmatrix} 1\\3 \end{bmatrix},$ it is clumsy to define a new variable $v_j$ that equals these constant numbers.

In summary, in an abstract vector space, we can easily use notation to distinguish the vector $v\in V$ itself and its components written as something like $ \begin{bmatrix} 1\\3 \end{bmatrix} $ in some basis. However, if we work with $V=\mathbb R^n,$ the distinction in notation between vector itself and components is lost.

Question: Are there any strategies to make such a distinction clear? What notation will prevent this ambiguity?

Answer 1

If $\beta=(\beta_1,\cdots,\beta_n)$ is an ordered basis of the vector space $V$ (over $\mathbb{R}$), and $v$ is an element of $V$ such that $$ v=v_1\beta_1+\cdots+v_n\beta_n $$ we write $[v]_\beta=(v_1,\cdots,v_n)$. (For convenience let me use row vectors.)

This notation works for both abstract $V$ and the concrete $\mathbb{R}^n$.

See for instance Linear Algebra by Friedberg-Insel-Spence.

---

Notes.

Consider $\mathbb{R}^2$ and the element $v=(1,1)$.

Under the standard basis $\beta=\{(1,0), (0,1)\}$, you have $$ [v]_\beta=[(1,1)]_{\beta}=(1,1) $$

Under the basis $\alpha=\{(2,0),(0,3)\}$, you have $$ [v]_\alpha=[(1,1)]_{\alpha}=(\frac 12,\frac13) $$