From S.L Linear Algebra:
Let $A$ be a non-zero vector in $R^2$. Let $F: \mathbb{R}^2
\rightarrow W$ be a linear map such that $F(A)=O$. Show that the image
of $F$ is either a straight line or $\{0\}$.
I've taken following theorems from the book to try and construct the answer (proofs of theorems are omitted):
Theorem 3.2. Let $V$ be a vector space. Let $L: V \rightarrow W$ be a linear map of $V$ into another space $W$. Let $n$ be the
dimension of $V$, $q$ the dimension of the kernel of $L$, and $s$ the
dimension of the image of $L$. Then $n = q + s$. In other words,$$\dim V= \dim \operatorname{Ker} L + \dim\operatorname{Im } L$$
Answer that I have constructed assumes all possibilities of dimension that kernel might have (due to cardinality, and Theorem 3.2, $\dim\operatorname{Ker}F \in \{0, 1, 2\}$ of a linear map $F$).
Possibility 1) $\dim\operatorname{Ker} F = 2$
If the dimension of kernel is $2$, that is, image is zero dimensional according to Theorem 3.2 ($\dim \operatorname{Im} F = \dim\mathbb{R}^{2} – \dim\operatorname{Ker}F = 2 – 2 = 0$), then considering that kernel is a subspace, $\mathbb{R}^2 = \operatorname{Ker}F$, and therefore $F$ is a zero map having the image of $\{0\}$.
Possibility 2) $\dim\operatorname{Ker}F = 1$
If kernel is $1$-dimensional, then so is the image according to Theorem 3.2 and thus we have a straight line as the image of $F$, considering that we have one-dimensional image under linear map $F$.
Possibility 3) $\dim\operatorname{Ker}F =\{0\} $ (Presumably impossible)
Kernel can't be zero-dimensional, since it is spanned by two dimensional vectors that are not zero vectors.
Conclusion:
Hence the image of $F$ is either a straight line or $\{0\}$.
Is this answer sufficient and true? My explanation of Possibility 2) concerns me the most, it might not be specific enough.
Thank you!
Best Answer
Your approach is correct!
P1) $\dim(Im \ F)=0 \implies Im(F)=\{0\}$, because the image of a linear function is a subspace and hence $0$ is in it, and can't have anything else because its dimension is zero. So $F(x)=0 \ \forall x$
P2) we have $\dim(Ker \ F)=1$, applying the theorem you get $\dim(Im \ T)=1$ and you can use the fact that two vector spaces are isomorphic (they are "the same space") if their dimension are equal, hence you can say that $Im(T)\cong \mathbb{R}$ which is a very nice way to justify that "$Im(T)$ is a straight line".
P3) can't be the case that $\dim(Ker \ T)=0$ because this would implie $Ker(T)=\{0\}$, but we know that $A\not=0$ and $A\in Ker(T)$
Your answer is good too! But it seems like it need to be more "direct" in a way... but the question isn't too direct either... I assumed that "being a straight line" is the same that "have dimension one"... but justifying that dimension one implies being isomorphic to the reals is also a good argument (because they are often called THE line).