What Does \Re2 \rightarrow \Re2 Mean in Linear Transformations?

[DmytriE]

Hi Pf,

Here is a question regard a test review that we have. I am not looking for the answer but rather a clarification about the notation.

1. What does the following mean? T₁: $\Re$² $\rightarrow$ $\Re$² by x $\rightarrow$ Ax?

2. What does it mean to go $\Re$² $\rightarrow$ $\Re$²

Thanks.

 

[chiro]

Hi Pf,

Here is a question regard a test review that we have. I am not looking for the answer but rather a clarification about the notation.

1. What does the following mean? T₁: $\Re$² $\rightarrow$ $\Re$² by x $\rightarrow$ Ax?

2. What does it mean to go $\Re$² $\rightarrow$ $\Re$²

Thanks.

Hey DmytriE.

Basically the LHS of the arrow is your starting space and the RHS is your target space. In other words we are starting in R^2 (2D vector with real numbers in each element) and we are going to a 2D vector.

In terms x -> Ax, this means that we start with a vector x and then we apply the operator A to the vector x by calculating Ax using normal matrix multiplication to get a new vector (still in R^2) called x' where x' = Ax.

 

[DmytriE]

Thanks Chiro!

 

[Fredrik]

The technical terms for "starting space" and "target space" are "domain" and "codomain". The notation $\mathbb R$ (\mathbb R) is more common than $\Re$ (\Re).

It strikes me as a bit odd to write "$T_1:\mathbb R^2\to\mathbb R^2$ by $x\to Ax$". The notation $T_1:\mathbb R^2\to\mathbb R^2$ tells us that $T_1$ is a function with domain $\mathbb R^2$ and codomain $\mathbb R^2$. The notation $x\to Ax$ should mean "the function that takes x to Ax". Wouldn't you denote that function by A? Hm, I guess that the most likely explanation is that A is a matrix, and the person who wrote this would like to emphasize that the linear operator T₁ that corresponds to the matrix A isn't the same thing as A (even though T₁ acting on x gives us the same result as A times x).

By the way, I prefer to use the \mapsto arrow in the second notation, i.e. I would write $x\mapsto Ax$ instead of $x\to Ax$. Some people prefer to never use the mapsto arrow. That's OK too.

 

[DmytriE]

I was trying to us the \mapsto arrow but when I clicked on it it gave me the bidirectional arrow and I did not know it's proper name so I just selected the normal arrow.

Here is another thing that confused me and my professor explained but I guess I was unable to grasp it. What is the difference between onto and one-to-one? What quantities would you look at to determine if it is one-to-one or onto (i.e. Null(A) or Col(A))? An example would help a lot.

 

[Fredrik]

A function f:X→Y is said to be injective (or one-to-one) if for all x,y in X, f(x)=f(y) implies x=y.

A function f:X→Y is said to be surjective (or onto) if for all y in Y, there's an x in X such that f(x)=y.

You can find examples in the wikipedia articles for these terms.