Dot product significance in vector transformation?

Question

Suppose we multiply a 3 component vector by some 3x3 transformation matrix. Is it correct, then, to say the following about the transformed vector?

Each component of the transformed vector is equal to the dot product of the original vector and the transformed basis vector corresponding to that component.

ex.. transformed x = v dot Mx, transformed y = v dot My, transformed z = v dot Mz

Which then leads to saying that the transformed vector's 3 components are the scaled cosines of the angle between the original vector and each of the transformation's basis vectors.

Drawing this out on paper, something doesn't seem quite right about this, but it looks like there's a connection. I would appreciate it if anyone could help me understand whether or not this is correct. Thanks.

Answer 1

So the question that you have seems to be whether it is necessarily the case that, letting $e_x,e_y,e_z$ denote the standard basis vectors, whether, for example: $$ M(v) \cdot e_x = v \cdot e_x $$ In fact, we can ask more generally: will it necessarily be the case that, for any two vectors $u$ and $v$, $$ M(u) \cdot v = u \cdot M(v)? $$ In fact, the answer is no: this will not be true for every matrix $M$. However, it is a commonly used fact that this will be true if and only if the matrix $M$ is symmetric. That is, we can say

for any matrix, $M = M^T$ if and only if for every vector $u,v,$ we have $M(u) \cdot v = u \cdot M(v)$, which is true if and only if for every $u,v \in \{e_x,e_y,e_z\}$, we have $M(u) \cdot v = u \cdot M(v)$.

In order to prove that this is the case, it is useful to notice that we can generally write $$ u \cdot v = v \cdot u = u^Tv $$