Quotient Spaces - Understanding Notation and Proving linear map

Question

Suppose $T\in L(V,W)$. Define $G: V/(kernel(T)) \to W$ by:

$G(\vec{v} + kernel(T)) = T(\vec{v})$

i) Show $G$ is a linear map

ii) Show $G$ is injective

iii) Show $range(T) = range(G)$

iv) Show $V/kernel(T)$ is isomorphic to $range(T)$

I am just getting started with the idea of a quotient space and understand that it is basically a vector space that has been translated but I am getting confused with what the notation is stating.

Can anyone provide an easier way to understand what G consists of and how to approach any of the questions below?

Answer 1

To make things easier to write let me abbreviate $K = \mathrm{ker}(T)$.

The notation $v + K$ means the subset of $V$ given by $$ v + K = \{ v+w ~|~ w \in K \} $$ By definition this whole set $v+K$ is just ONE ELEMENT of the set $V/K$. That means the elements of $V/K$ are sets of the form $v+K$ for different choices of $v$.

Say you have $w \in K$. Then $$ T(v+w) = T(v)+T(w) = T(v) + 0 = T(v) $$ What this computation shows is that ALL elements of $v + K$ map to the same point under $T$. The definition of $G$ is that it sends the element $(v+K) \in V/K$ to this point (the common image under $T$ of all the points in $v+K$).

I hope this clarifies the notation.

Answer 2

In elementary school we're taught that division can be thought of as reorganizing some set of objects into subgroups. For example, the operation of dividing $4$ by $2$ can be seen as taking $4$ objects and organizing them into groups with the property they have $2$ elements.

Quotient spaces follow the same idea, where we take all the vectors in a vector space and reorganize them into equivalence classes according to some property. In this case, an element of $V/(\text{kernel}(T))$ is a coset associated with some vector $r\in V$ such that any vector in this coset can be expressed as $r+k$ where $k\in \text{kernel}(T)$. An even better way to describe $V/(\text{kernel}(T))$ is as a set of cosets where vectors within the same coset get mapped to the same vector by $T$. This is because any vector in $V$ can be expressed as the sum of a component lying in the kernel space and another vector that is orthogonal to the kernel space.

As an example, consider the case that $T$ can be written as left matrix multiplication. From rank-nullity theorem and the fact row-rank = column-rank any $F: V/(\text{kernel}(T))\rightarrow W$ can be seen as a function whose domain is the projection of $V$ onto the row-space of $T$.

Now onto your questions. As mentioned earlier, a coset $C$ in this case has the property that every vector in $C$ gets mapped to the same output vector by $T$. The value of $G(C)$ is simply the output vector in question.

i) The cosets $C_1$ and $C_2$ can be represented by vectors $r_1$ and $r_2$ so $aC_1+bC_2$ can be represented by the vector $ar_1+br_2$ and $$G(aC_1+bC_2)=T(ar_1+br_2)=aT(r_1)+bT(r_2)=aG(C_1)+bG(C_2)$$

ii) $G$ is injective because if any two vectors are mapped to the same place by $T$, then these two vectors are in the same coset.

iii) Follows from the observation $G(C)=T(v)$ for any $v\in C$

iv) Consequence of i)-iii)

If anything is confusing, feel free to let me know and I will try and clarify.