Defining Transformations given a set of elements (Apostol Volume 2)
$begingroup$
The question is laid out like this:
Let $V = {0,1}$ . Describe all functions $T: Vlongrightarrow V$ . There are four altogether. Label them as
$T_1 , T_2 , T_3, T_4$ and make a multiplication table showing the composition of each pair. Indicate
which functions are one-to-one on $V$ and give their inverses.
Since it is giving us the elements of $V$, I am not quite sure how to get the problem going. I was initially thinking the identity transformation and Zero transformation, and scalar, but the question also doesn't specify if it is linear so if it transforms every element to $1$ persay that will still be in $V$, but it wont preserve addition, or if were need the Zero of $V$ to map back to the Zero as earlier in the section he showed "Right" inverses mapping Zero's from $W$ back to scalars in $V$.
Then I looked at the next question which says:
Let $V = {0,1,2}$. Describe all functions $T: Vlongrightarrow V$ for which $T(V)=V$. There are six altogether. Label them as $T_1, . . . , T_6$ and make a multiplication table showing the composition of each pair. Indicate which functions are one-to-one on $V$, and give their inverses.
So $V$ isn't even necessarily a basis (I was going to try something with polynomials). So I really don't know how to get this one rolling.
linear-algebra transformation
edited Jan 16 at 12:07
idriskameni
746321
asked Jan 16 at 7:26
ARichardsonARichardson
12
$endgroup$
add a comment |
$begingroup$
The question is laid out like this:
Let $V = {0,1}$ . Describe all functions $T: Vlongrightarrow V$ . There are four altogether. Label them as
$T_1 , T_2 , T_3, T_4$ and make a multiplication table showing the composition of each pair. Indicate
which functions are one-to-one on $V$ and give their inverses.
Since it is giving us the elements of $V$, I am not quite sure how to get the problem going. I was initially thinking the identity transformation and Zero transformation, and scalar, but the question also doesn't specify if it is linear so if it transforms every element to $1$ persay that will still be in $V$, but it wont preserve addition, or if were need the Zero of $V$ to map back to the Zero as earlier in the section he showed "Right" inverses mapping Zero's from $W$ back to scalars in $V$.
Then I looked at the next question which says:
Let $V = {0,1,2}$. Describe all functions $T: Vlongrightarrow V$ for which $T(V)=V$. There are six altogether. Label them as $T_1, . . . , T_6$ and make a multiplication table showing the composition of each pair. Indicate which functions are one-to-one on $V$, and give their inverses.
So $V$ isn't even necessarily a basis (I was going to try something with polynomials). So I really don't know how to get this one rolling.
linear-algebra transformation
edited Jan 16 at 12:07
idriskameni
746321
asked Jan 16 at 7:26
ARichardsonARichardson
12
$endgroup$
add a comment |
$begingroup$
The question is laid out like this:
Let $V = {0,1}$ . Describe all functions $T: Vlongrightarrow V$ . There are four altogether. Label them as
$T_1 , T_2 , T_3, T_4$ and make a multiplication table showing the composition of each pair. Indicate
which functions are one-to-one on $V$ and give their inverses.
Since it is giving us the elements of $V$, I am not quite sure how to get the problem going. I was initially thinking the identity transformation and Zero transformation, and scalar, but the question also doesn't specify if it is linear so if it transforms every element to $1$ persay that will still be in $V$, but it wont preserve addition, or if were need the Zero of $V$ to map back to the Zero as earlier in the section he showed "Right" inverses mapping Zero's from $W$ back to scalars in $V$.
Then I looked at the next question which says:
Let $V = {0,1,2}$. Describe all functions $T: Vlongrightarrow V$ for which $T(V)=V$. There are six altogether. Label them as $T_1, . . . , T_6$ and make a multiplication table showing the composition of each pair. Indicate which functions are one-to-one on $V$, and give their inverses.
So $V$ isn't even necessarily a basis (I was going to try something with polynomials). So I really don't know how to get this one rolling.
linear-algebra transformation
edited Jan 16 at 12:07
idriskameni
746321
asked Jan 16 at 7:26
ARichardsonARichardson
12
$endgroup$
The question is laid out like this:
Let $V = {0,1}$ . Describe all functions $T: Vlongrightarrow V$ . There are four altogether. Label them as
$T_1 , T_2 , T_3, T_4$ and make a multiplication table showing the composition of each pair. Indicate
which functions are one-to-one on $V$ and give their inverses.
Since it is giving us the elements of $V$, I am not quite sure how to get the problem going. I was initially thinking the identity transformation and Zero transformation, and scalar, but the question also doesn't specify if it is linear so if it transforms every element to $1$ persay that will still be in $V$, but it wont preserve addition, or if were need the Zero of $V$ to map back to the Zero as earlier in the section he showed "Right" inverses mapping Zero's from $W$ back to scalars in $V$.
Then I looked at the next question which says:
Let $V = {0,1,2}$. Describe all functions $T: Vlongrightarrow V$ for which $T(V)=V$. There are six altogether. Label them as $T_1, . . . , T_6$ and make a multiplication table showing the composition of each pair. Indicate which functions are one-to-one on $V$, and give their inverses.
So $V$ isn't even necessarily a basis (I was going to try something with polynomials). So I really don't know how to get this one rolling.
linear-algebra transformation
linear-algebra transformation
edited Jan 16 at 12:07
idriskameni
746321
asked Jan 16 at 7:26
ARichardsonARichardson
12
edited Jan 16 at 12:07
idriskameni
746321
asked Jan 16 at 7:26
ARichardsonARichardson
12
edited Jan 16 at 12:07
idriskameni
746321
edited Jan 16 at 12:07
idriskameni
746321
edited Jan 16 at 12:07
idriskameni
746321
746321
asked Jan 16 at 7:26
ARichardsonARichardson
12
asked Jan 16 at 7:26
ARichardsonARichardson
12
asked Jan 16 at 7:26
ARichardsonARichardson
12
12
add a comment |
add a comment |
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