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

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?

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 $$[v]_\alpha=[(1,1)]_{\alpha}=(\frac 12,\frac13)$$