Definitions of Functions and Spaces

[imfromyemen]

Hi everyone,
I am in second year university and am taking linear algebra this semester. Never having been a strong maths student, I am certainly struggling with some basic concepts and especially notation.

I have tried searching on the web but have had difficulty in finding something which properly explains the meaning of notation like

$$ f: \Bbb{R^2} \to \Bbb{R}$$ or the difference between
$x\in \Bbb{R^n}$ and $x \in \Bbb{R}$I can basically read these, and know the literal pronounciation of the symbols, but have no idea what they actually mean.

The first one would be $f$ maps $\Bbb{R^2}$ to $\Bbb{R}$. What does this mean exactly?

Is it saying that on an (x,y) plane, the function f returns a single point?
so for example - f(x) = 4x, then f(1) = 4
is the second one saying that x is an element of a vector space with n elements $(ax_1, bx_2,...,a_nx_n)$, whereas the first one is saying that x is just some real number?

I would really appreciate if someone could help me with this, I want to do really well in mathematics and not being able to understand these little things is making everything much more difficult than it need be.

kind regards

 

[mathmari]

I have tried searching on the web but have had difficulty in finding something which properly explains the meaning of notation like

$$ f: \Bbb{R^2} \to \Bbb{R}$$ or the difference between
$x\in \Bbb{R^n}$ and $x \in \Bbb{R}$I can basically read these, and know the literal pronounciation of the symbols, but have no idea what they actually mean.

The first one would be $f$ maps $\Bbb{R^2}$ to $\Bbb{R}$. What does this mean exactly?

Is it saying that on an (x,y) plane, the function f returns a single point?
so for example - f(x) = 4x, then f(1) = 4
is the second one saying that x is an element of a vector space with n elements $(ax_1, bx_2,...,a_nx_n)$, whereas the first one is saying that x is just some real number?

I would really appreciate if someone could help me with this, I want to do really well in mathematics and not being able to understand these little things is making everything much more difficult than it need be.

kind regards

Hello! (Smile)

The notation $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ means that $f$ is a function whose domain is $\mathbb{R}^2$, the set of real $2-$vectors, and whose range is some subset of $\mathbb{R}$, possibly but not necessarily all of $\mathbb{R}$.

An example of such a function is the following: $f(x, y)=x^2y$. $\mathbb{R}^n$ is the set of ordered $n-$tuples of real numbers, $$\mathbb{R}^n=\{(x_1, x_2, \dots , x_n): x_1, x_2, \dots , x_n \in \mathbb{R}\}$$

The elements of $\mathbb{R}^n$ are called vectors. Given a vector $x=(x_1, x_2 \dots , x_n)$, the numbers $x_1, x_2 , \dots , x_n$ are called components of $x$.


At the definition of $\mathbb{R}^n$, for $n=1$ we get $\mathbb{R}^1=\mathbb{R}$ which can be interpreted as the number line.

In other words, $\mathbb{R}$ is the set of real numbers.