Mixing
Question about Image and Kernel in a combined Linear Transformation
$begingroup$
I am new to Linear Algebra, and was asked the following question:
Let $S,T:Vto V$ be linear transformations, while V is a linear space.
If $mbox{im} ST={0}$ then $mbox{im} Tsubseteq ker S$
I have tried to reason this through, but I am not sure whether my reasoning is correct:
$$ker S: Sunderline x=0$$
$$mbox{im} T = text{ column-space of }T$$
So if I multiply $S times T$, with the result being ${0}$, that would seem to imply that indeed $T$ is a subset of $ker S$, as it obeys $Sunderline x=0$, with $T$ taking the place of $underline x$.
Can anybody help out?
Thank you
linear-algebra linear-transformations
edited Jan 1 at 16:44
amWhy
192k28225439
asked Jan 1 at 16:04
daltadalta
627
$endgroup$
add a comment |
$begingroup$
I am new to Linear Algebra, and was asked the following question:
Let $S,T:Vto V$ be linear transformations, while V is a linear space.
If $mbox{im} ST={0}$ then $mbox{im} Tsubseteq ker S$
I have tried to reason this through, but I am not sure whether my reasoning is correct:
$$ker S: Sunderline x=0$$
$$mbox{im} T = text{ column-space of }T$$
So if I multiply $S times T$, with the result being ${0}$, that would seem to imply that indeed $T$ is a subset of $ker S$, as it obeys $Sunderline x=0$, with $T$ taking the place of $underline x$.
Can anybody help out?
Thank you
linear-algebra linear-transformations
edited Jan 1 at 16:44
amWhy
192k28225439
asked Jan 1 at 16:04
daltadalta
627
$endgroup$
$begingroup$
Shouldn't it be the composition of linear maps $S$ and $T$ rather than mentioning it as multiplication? @dalta
$endgroup$
– toric_actions
Jan 1 at 16:17
add a comment |
$begingroup$
I am new to Linear Algebra, and was asked the following question:
Let $S,T:Vto V$ be linear transformations, while V is a linear space.
If $mbox{im} ST={0}$ then $mbox{im} Tsubseteq ker S$
I have tried to reason this through, but I am not sure whether my reasoning is correct:
$$ker S: Sunderline x=0$$
$$mbox{im} T = text{ column-space of }T$$
So if I multiply $S times T$, with the result being ${0}$, that would seem to imply that indeed $T$ is a subset of $ker S$, as it obeys $Sunderline x=0$, with $T$ taking the place of $underline x$.
Can anybody help out?
Thank you
linear-algebra linear-transformations
edited Jan 1 at 16:44
amWhy
192k28225439
asked Jan 1 at 16:04
daltadalta
627
$endgroup$
I am new to Linear Algebra, and was asked the following question:
Let $S,T:Vto V$ be linear transformations, while V is a linear space.
If $mbox{im} ST={0}$ then $mbox{im} Tsubseteq ker S$
I have tried to reason this through, but I am not sure whether my reasoning is correct:
$$ker S: Sunderline x=0$$
$$mbox{im} T = text{ column-space of }T$$
So if I multiply $S times T$, with the result being ${0}$, that would seem to imply that indeed $T$ is a subset of $ker S$, as it obeys $Sunderline x=0$, with $T$ taking the place of $underline x$.
Can anybody help out?
Thank you
linear-algebra linear-transformations
linear-algebra linear-transformations
edited Jan 1 at 16:44
amWhy
192k28225439
asked Jan 1 at 16:04
daltadalta
627
edited Jan 1 at 16:44
amWhy
192k28225439
asked Jan 1 at 16:04
daltadalta
627
edited Jan 1 at 16:44
amWhy
192k28225439
edited Jan 1 at 16:44
amWhy
192k28225439
edited Jan 1 at 16:44
amWhy
192k28225439
192k28225439
asked Jan 1 at 16:04
daltadalta
627
asked Jan 1 at 16:04
daltadalta
627
asked Jan 1 at 16:04
daltadalta
627
627
$begingroup$
Shouldn't it be the composition of linear maps $S$ and $T$ rather than mentioning it as multiplication? @dalta
$endgroup$
– toric_actions
Jan 1 at 16:17
add a comment |
$begingroup$
Shouldn't it be the composition of linear maps $S$ and $T$ rather than mentioning it as multiplication? @dalta
$endgroup$
– toric_actions
Jan 1 at 16:17
$begingroup$
Shouldn't it be the composition of linear maps $S$ and $T$ rather than mentioning it as multiplication? @dalta
$endgroup$
– toric_actions
Jan 1 at 16:17
$begingroup$
Shouldn't it be the composition of linear maps $S$ and $T$ rather than mentioning it as multiplication? @dalta
$endgroup$
– toric_actions
Jan 1 at 16:17
add a comment |
1 Answer
1
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oldest
votes
$begingroup$
I think you have the right idea - let's just be slightly more precise.
Suppose $y in textrm{Im}(T)$, so $y = T(x)$ for some $x in V$. Then $S(y) = ST(x) = 0$ since $ST = 0$ and hence $y in textrm{Ker}(S)$. Therefore $textrm{Im}(T) subseteq textrm{Ker}(S)$.
answered Jan 1 at 16:11
ODFODF
1,396510
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add a comment |
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
I think you have the right idea - let's just be slightly more precise.
Suppose $y in textrm{Im}(T)$, so $y = T(x)$ for some $x in V$. Then $S(y) = ST(x) = 0$ since $ST = 0$ and hence $y in textrm{Ker}(S)$. Therefore $textrm{Im}(T) subseteq textrm{Ker}(S)$.
answered Jan 1 at 16:11
ODFODF
1,396510
$endgroup$
add a comment |
$begingroup$
I think you have the right idea - let's just be slightly more precise.
Suppose $y in textrm{Im}(T)$, so $y = T(x)$ for some $x in V$. Then $S(y) = ST(x) = 0$ since $ST = 0$ and hence $y in textrm{Ker}(S)$. Therefore $textrm{Im}(T) subseteq textrm{Ker}(S)$.
answered Jan 1 at 16:11
ODFODF
1,396510
$endgroup$
add a comment |
$begingroup$
I think you have the right idea - let's just be slightly more precise.
Suppose $y in textrm{Im}(T)$, so $y = T(x)$ for some $x in V$. Then $S(y) = ST(x) = 0$ since $ST = 0$ and hence $y in textrm{Ker}(S)$. Therefore $textrm{Im}(T) subseteq textrm{Ker}(S)$.
answered Jan 1 at 16:11
ODFODF
1,396510
$endgroup$
I think you have the right idea - let's just be slightly more precise.
Suppose $y in textrm{Im}(T)$, so $y = T(x)$ for some $x in V$. Then $S(y) = ST(x) = 0$ since $ST = 0$ and hence $y in textrm{Ker}(S)$. Therefore $textrm{Im}(T) subseteq textrm{Ker}(S)$.
answered Jan 1 at 16:11
ODFODF
1,396510
answered Jan 1 at 16:11
ODFODF
1,396510
answered Jan 1 at 16:11
ODFODF
1,396510
answered Jan 1 at 16:11
ODFODF
1,396510
1,396510
add a comment |
add a comment |
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$begingroup$
Shouldn't it be the composition of linear maps $S$ and $T$ rather than mentioning it as multiplication? @dalta
$endgroup$
– toric_actions
Jan 1 at 16:17