Confused about vectors and transformations (linear)

[GiuseppeR7]

when we are talking about a linear transformation the argument of the function is a coordinate vector...is this true?
another question...when i see a column vector...these are the coordinates of the vector with respect of a basis...is this true? for example if i see...

$(({{1},{3}}))^T$

with respect to a basis ${a1,a2}$...that "column symbol" are the coordinates of this vector?:
$v=a1*1+a2*3$
??

 

[Stephen Tashi]

when we are talking about a linear transformation the argument of the function is a coordinate vector...is this true?

Abstractly, one can talk about a linear transformation as a function whose argument is a single symbol, such as $T(X)$ where $X$ is a vector. Linearity requirements such as $T(\alpha X + Y) = \alpha T(X) + T(Y)$ can be stated without reference to particular coordinates. (For example, in 2-D polar coordinates, the above requirement can be implemented by a coordinate operations that are different from an implementation in cartesian coordinate operations.)

another question...when i see a column vector...these are the coordinates of the vector with respect of a basis...is this true? for example if i see...

$(({{1},{3}}))^T$

with respect to a basis ${a1,a2}$...that "column symbol" are the coordinates of this vector?:
$v=a1*1+a2*3$
??

I'd say yes.

When a vector is expressed in terms of a linear combination of a particular (finite) set of basis vectors then a customary way to represent a vector $X$ is as a column vector whose entries are the scalars in the linear combination. (So it is fair to call that representation a "coordinate representation", although it does not necessarily refer to coordinates Euclidean space.) Then a linear transformation $T$ amounts to left multiplying the column vector by a matrix of constants.

 

[GiuseppeR7]

ok, thanks a lot!