# What is the transformation matrix "by definition"?

I've taken a course in linear algebra two semesters ago and we have talked about linear transformations and matrices. Matrices can represent linear transformations.

Today in our first lecture of computational science, the professor started by recalling what's a linear transformation and what's a matrix, and he said that a matrix is not a linear transformation, but it is a representation of a linear transformation with respect to a basis, which I agreed, but I'm not convinced 100%.

Could you please provide a full and self-contained explanation of what my professor said in class today?

The second question that I have is related to this video lecture on you:

At minute 2:10, the video lecturer says that the matrix

$$\begin{pmatrix} 1 & 1 & 0 \\ 1 & -1 & 0 \\0 & 0 & 3\end{pmatrix}$$

defines the linear transformation

$$\begin{pmatrix} a \\ b \\c\end{pmatrix} \rightarrow \begin{pmatrix} a + b \\ a- b \\ 3c\end{pmatrix}$$

which, IMO, it is a little bit contradictory to what my professor said today during the lecture. He says that that matrix is the definition.

I think the video lecturer is wrong or at least hasn't explained well the concept. I think he should have said that this matrix represents the linear transformation with respect to the standard basis, i.e.

$$\begin{pmatrix} 1 \\ 0 \\0\end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\0\end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\1\end{pmatrix}$$

but I would not know exactly how to prove I'm correct or wrong, since I've forgotten many things from the course I'd taken in linear algebra almost a year ago.

• a) "defines" is not "is". And b) the linear transformation defined by a matrix is between very specific vector spaces, here from $\Bbb R^3$ to $\Bbb R^3$ and not, say, from the space of solutions of the differential equation $y'''+2y=0$ to some other space without a natural basis. - But your doubts are not wrong Feb 22, 2016 at 22:31
• Your ideas are correct. You seem to know what you're talking about. Feb 22, 2016 at 22:37

A matrix, e.g. $A$, is just a bunch of numbers arranged into a rectangular shape, while a transformation, e.g. $T$, is a map between two sets.

These are different things.

However, it is possible to represent a linear transformation by a matrix. So there exists a mapping between both. Given a matrix one can come up with the linear transformation and vice versa.

The idea is to note how the transformation acts on the base vectors: $$x' = T(x) = T(x_i \,e_i) = x_i T(e_i) = x_i b_{ij} e_j = b_{ij} x_i e_j$$ where we sum over same index variables (Einstein summation convention) or $$x'_i = a_{ij} x_j$$ with $A = B^{-1}$, $A =(a_{ij})$, $B = (b_{ij})$.

Thus the complication is that one has to chose a basis, or coordinate system, to come up with a particular matrix for a given linear transformation.

If nothing is said about the chosen basis one can assume the canonical basis is used. (Another convention)

Seems we agree.