Understanding Singular Linear Maps: R^m -> R^n

[annoymage]

Homework Statement

the question here said

is L, linear transformation/mapping is singular?

i'm still googling the definition singular linear map,

can anyone give me the definition please T_T

p/s; i thought it L maybe the matrix representation, but the question

L : R^m -> R^n

but aren't matrix representation are define on L: M(m,1) -> M(n,1)

should i take transpose of R^m and R^n ?

 

[jbunniii]

If L is a linear map between finite-dimensional vector spaces (such as $\mathbb{R}^m$ and $\mathbb{R}^n$), then it can be represented by a matrix. In fact, it can be represented by many different matrices, each corresponding to different choices of basis. But sometimes the answer to a question can be clearer if you don't focus on the matrix representation. Unfortunately, you did not fully state the question! (i.e., how is L defined?)

In any case, a singular linear map is simply one that does not have an inverse. It can fail to have an inverse in one (or both) of two ways: (1) it is not injective, meaning there exist $a \neq b \in \mathbb{R}^m$ such that $L(a) = L(b)$; or (2) it is not surjective, meaning that there exists some $b \in \mathbb{R}^n$ such that $L(a) \neq b$ for all $a \in \mathbb{R}^m$.

If $m \neq n$, then $L$ is guaranteed to be singular. (Why?)

 

[annoymage]

thanks for the definition.

And your question, I can see If m<n then not surjective means singular

if m>n then not injective means singular. But still working on that. Tomorrow i'll try to post, help check it.

Anyway, i don't understand this,

If L is a linear map between finite-dimensional vector spaces (such as $\mathbb{R}^m$ and $\mathbb{R}^n$), then it can be represented by a matrix. In fact, it can be represented by many different matrices, each corresponding to different choices of basis.

what topic should i study for this thing?

 

[jbunniii]

And your question, I can see If m<n then not surjective means singular

Correct, surjectivity is impossible when $m < n$, and injectivity is impossible when $m > n$.

Anyway, i don't understand this

what topic should i study for this thing?

Well, the topic is linear algebra, but you can study it at several levels of sophistication. Are you taking a course in it? If so, have you seen the definition of a linear map between two vector spaces? It should look something like this:

If V and W are vector spaces over the same field K and $L : V \rightarrow W$, then L is a linear map if the following are true for all $v, v_1, v_2 \in V$, and $k \in K$:

(1) $L(v_1 + v_2) = L(v_1) + L(v_2)$

(2) $L(kv) = kL(v)$

 

[annoymage]

what topic should i study for this thing?

doesn't matter, i think i found it, i'll read it forthwith

 

[annoymage]

Well, the topic is linear algebra, but you can study it at several levels of sophistication. Are you taking a course in it? If so, have you seen the definition of a linear map between two vector spaces? It should look something like this:

If V and W are vector spaces over the same field K and $L : V \rightarrow W$, then L is a linear map if the following are true for all $v, v_1, v_2 \in V$, and $k \in K$:

(1) $L(v_1 + v_2) = L(v_1) + L(v_2)$

(2) $L(kv) = kL(v)$

yea i did pass that topic, only my lecture note doesn't define what singularity of linear map is, anyway thank you so much