The correct way to write the range of a linear transformation

[Hall]

Homework Statement

Find the range of T.

Relevant Equations

Range of T is written as $T(V_2)$.

We have a transformation $T : V_2 \to V_2$ such that:
$$
T (x,y)= (x,x)
$$

Prove that the transformation is linear and find its range.

We can prove that the transformation is Linear quite easily. But the range $T(V_2)$ is the the line $y=x$ in a two dimensional (geometrically) space. Is that the correct way to write the range of T?

For another case, if $T(x,y) = (x,0)$ then the range $T(V_2)$ will be the x-axis. But I think they need to be written more symbolically rather than saying "the line y=x is the range" or "the x-axis is the range".

 

[PeroK]

Do you know set notation?$$S = \{X \in U: P(X) \}$$The set $S$ is defined as a subset of some universal set $U$, where the members of $S$, denoted by $X$, have the property $P(X)$

 

[Hall]

Do you know set notation?$$S = \{X \in U: P(X) \}$$The set $S$ is defined as a subset of some universal set $U$, where the members of $S$, denoted by $X$, have the property $P(X)$

Okay.
$$
T(V_2 )= \{ (x,x) | x \in \mathcal{R}\}
$$

 

[PeroK]

Okay.
$$
T(V_2 )= \{ (x,x) | x \in \mathcal{R}\}
$$

Is $V_2$ the Cartesian product of $V$ with itself? In that case, it's $x \in V$.

 

[Hall]

Is $V_2$ the Cartesian product of $V$ with itself? In that case, it's $x \in V$.

I had two basic questions -- having a new thread for that doesn't seem the need--
1. Can a singleton set be a Linear space?
2. Is an empty set a Linear space?

 

[PeroK]

I had two basic questions -- having a new thread for that doesn't seem the need--
1. Can a singleton set be a Linear space?
2. Is an empty set a Linear space?

1) Yes. 2) No.

Look at the definition.

Can you think of an example of a linear space that is often the singleton $\{0\}$?

Why cannot $\emptyset$ be a linear space? What property does it lack?

 

[Hall]

Can you think of an example of a linear space that is often the singleton {0}?

Amm... yes. Every set which contains the zero element only, is a singleton set, satisfies all the axioms of a linear space. Examples: $\{(0,0)\}$, the set containing the zero polynomial, et cetra.

Why cannot ∅ be a linear space? What property does it lack?

I just re-looked at the definition of linear space and there it was written "A nonempty set V is called a linear space if..."

Can you please tell me why the dimension of a linear space, which contains the zero element only, is 0? I mean that space can be spanned by the zero element itself, can't it be?

 

[PeroK]

Can you please tell me why the dimension of a linear space, which contains the zero element only, is 0? I mean that space can be spanned by the zero element itself, can't it be?

It's similar to defining a point as zero dimensional. The dimension is zero by definition. Otherwise it would mess up counting dimensions.

You still haven't found an important example of why we must allow the zero vector to be a linear space?

What aspect of linear algebra theory would be messed up by disallowing the zero dimensional space?

 

[PeroK]

The dimension is zero by definition.

See Definition 1 here, for example:

https://sites.lafayette.edu/thompsmc/files/2015/08/Section_4_51.pdf

 

[Hall]

You still haven't found an important example of why we must allow the zero vector to be a linear space?

I mean, the closure axioms are followed and the set containing the zero element is a subset of a Linear space, hence it is a subspace.

 

[Hall]

What aspect of linear algebra theory would be messed up by disallowing the zero dimensional space?

The most recent I have found is :

Nullity + Rank = dimension of domain

Will be violated.

 

[PeroK]

The most recent I have found is :

Nullity + Rank = dimension of domain

Will be violated.

In other words, the kernel of a linear transformation is a subspace which often has dimension zero.