1

Given that the bases of a vector and their duals satisfy: $${\epsilon}^ie_j={\delta}^i_j$$ in which $\epsilon$ and $e$ are the dual bases and bases respectively, and ${\delta}^i_j$ is the Kronecker symbol, and that the definition of the dual vector space is: $$V^*=\{{\varphi}:V \longrightarrow \mathbb{R}\}$$ with $V$ being a vector space. Is it "correct" to think of the covector and vector as analogous to row and column vectors respectively ? I came to this conclusion due to the fact that the matrix product of a row vector and a column vector is a scalar, and $\varphi$ also takes vectors to numbers, but I have a feeling that I'm still missing something.

Thanks in advance!

P/s: I also find this way of thinking of vectors and covectors very nice since row and column vectors are sort of doppelgängers, hence the covectors and dual spaces

user518704
  • 73

1 Answers1

2

It is partially correct, or at least meaningful. Indeed, given $v \in V$ and $w \in V^*$, if you choose dual bases, you have that $w(v)=w^Tv$ (check this). But note that I am considering all vectors as vertical and then transposing $w$, since in principle $V^*$ is not a special vector space, so that it is not very precise to write its vectors in a different way.

57Jimmy
  • 6,266