The word function in calculus refers to something like $f(x) = x^2+2x^3$ or $f(x) =\sin(x) $ etc….
In linear algebra, the word function is used like-
A linear transformation is a function from $V \rightarrow W$.
And the functions of calculus like $f(x) = x^2+2x^3$ or $f(x) =\sin(x) $ etc. are actually vectors in either a polynomial space ($f(x) = x^2+2x^3$ ) or a Function space ( like $f(x) =\sin(x) $ ) .
Now the word function in Linear Algebra is used twice as I showed above.
So according to me the functions of calculus are just vectors in linear algebra. Is this correct or not.
But then what are the functions being used in the definition of Linear Transformations. And how are they different from the functions of calculus and the functions which are vectors in linear algebra.
Edit:
Why is the graph of a linear transformation from any vector space to any other vector space not always a straight line. Can anyone give any counter examples.
Best Answer
A function is defined as a relation between two sets that maps one element from one set to exactly one of the other set. For your example $f(x) = x^{2} + 2x^{3}$, the element $x$ in the domain is mapped to the codomain element $x^{2} + 2x^{3}$.
In your linear algebra example, your domain is denoted $V$ and your codomain is denoted $W$.
A linear transformation is one specific type of function where an extra restriction is required: $f(cx + y) = cf(x) + f(y)$. Both are examples of functions but this restriction placed on linear maps may or may not hold for functions in general.
Linear transformations can be graphed but they are commonly graphed as vector fields; a linear transformation graph would not look like your typical one-to-one function from calculus.