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Recently I got confused about the difference between a vector and a coordinate and here is what I concluded.

Given a vector space $V$ over a field $F$, a vector in the vector space $\vec{w}$ and a basis $\{\vec{v_1}, \vec{v_2}, ..., \vec{v_n}\}$ we can represent the vector as a unique linear combination of the basis vectors $$\vec{w} = a_1\vec{v_1} + a_2\vec{v_2}+...+a_n\vec{v_n}$$ then the scalars can form a tuple in the space $F^k$ with $k \leq{n}$ that tuple is what we call as the coordinate of the corresponding vector for the given basis.

Q1. is this description correct so far?

Now I recall how we have derived the fact that linear transformations can be represented as matrix-vector multiplication. Hubbard and Hubbard page 59

(Hubbard and Hubbard page 59, I hope its alright to just insert an image from the book itself) however, see that in the above derivation $v_i$ are just the scalars, (i.e. elements of the coordinate) in the standard basis and thus the tuple formed of $v_i$ on the right (the second last step) is just the coordinate representation of the vector and not the vector itself.

Q2. Does the matrix actually act on the coordinate but not the vector?

To make the matter clearer, I can take another example where we take the vector space $\mathbb{C}$ with field $\mathbb{R}$ over it with usual addition and scalar multiplication. However, I have one doubt do the basis vectors even exist in such a vector space because as far as i know, basis vectors are vectors under which the coordinate representation is identical to the vector itself as given in this answer but that can't seem to be the possible case here.

Q3. when do standard basis vectors exist?

Still, I presume that I can do the derivation given above if I have used any other basis than the standard basis. in the complex plane, I choose the basis vectors $\vec{1}$ and $\vec{i}$ and compose a matrix [M] as a linear transformation where $\vec{1}$ lands at $\vec{1+3i}$ and $\vec{i}$ lands at $\vec{3+2i}$. see that i cant do the matrix-vector multiplication of [M] with some vector in complex plane say $\vec{1+i}$ because that wont mean anything.

Q4. Is what i am saying correct? because it seems totally bizarre that i have been doing matrix coordinate multiplication this whole time for the linear transformation of vectors and there is no description of this anywhere.

Any additional comments and insights are welcome

Gaurang Agrawal
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  • Um, there is a definition of vector-vector product for complex numbers and such though, remember $i \cdot i = -1$ for example? I don't follow that point. – Emil Jan 09 '22 at 08:46
  • there is a definition that is used generally but we have not stated what we are using explicitly yet. just like how we have stated the usual scalar multiplication and usual vector addition. we could have as well stated another definition say by scalar multiplication $a*\vec{v}$ we mean $2a\vec{v}$ and it would work. but for now we don't have what we mean by vector vector product – Gaurang Agrawal Jan 09 '22 at 08:56
  • Then it is your "1" and "i" space, totally unrelated to the complex plane. – Emil Jan 09 '22 at 09:22
  • Did you mean two-dimensional vector space over complex numbers or did you mean the complex plane (complex numbers represented as two dimensional algebra/vector space over reals with a vector-vector product) – Emil Jan 09 '22 at 09:29
  • its complex plane and i got it we need to define the product of vectos in complex plane say $i\cdot i$ i will correct that. – Gaurang Agrawal Jan 09 '22 at 09:53
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    "No description of it anywhere" is a bit strong, though I agree that there's too much hand-waving in many accounts. I think this is clear though https://math.libretexts.org/Bookshelves/Linear_Algebra/Book%3A_Linear_Algebra_(Schilling_Nachtergaele_and_Lankham)/06%3A_Linear_Maps/6.06%3A_The_matrix_of_a_linear_map – ancient mathematician Jan 09 '22 at 10:11
  • @ancient yeah you are right. the book I was using ( hubbard and hubbard) didn't state it explicitly and most of the places i go they use the 'matrix-vector' product. while i see that the elements of coordinate space F^n are also vectors one should be clear that while doing [M]v v doesn't lie in vector space but in coordinate space. I was confused when I encountered this while dealing with finite polynomial spaces Pn for which the author suddenly started using matrix coordinate product. – Gaurang Agrawal Jan 09 '22 at 10:27

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