Mixing
Easier way to calculate Nullity of triple composition of a linear transformation?
$begingroup$
$T:R^3to R^3$
$T(x,y,z)=(x+3y+2z,3x+4y+z,2x+y-z)$
I want to find the dimension of the Null Space of $T^3$
Here Nullity of $T$ is not $0$, so I can't say anything directly about the nullity of $T^3$
Normally for the nullity of T, I would form the matrix of transformation by take the image of the standard basis of $R^3$, then row reduce to calculate the rank and use the dimension theorem.
But doing the same for $T^3$ is turning out to be hefty task and I am looking for a easier way.
linear-algebra vector-spaces linear-transformations
asked Jan 29 at 6:32
AbhayAbhay
3789
$endgroup$
add a comment |
$begingroup$
$T:R^3to R^3$
$T(x,y,z)=(x+3y+2z,3x+4y+z,2x+y-z)$
I want to find the dimension of the Null Space of $T^3$
Here Nullity of $T$ is not $0$, so I can't say anything directly about the nullity of $T^3$
Normally for the nullity of T, I would form the matrix of transformation by take the image of the standard basis of $R^3$, then row reduce to calculate the rank and use the dimension theorem.
But doing the same for $T^3$ is turning out to be hefty task and I am looking for a easier way.
linear-algebra vector-spaces linear-transformations
asked Jan 29 at 6:32
AbhayAbhay
3789
$endgroup$
$begingroup$
Are you calculating the matrix $T^3$ or trying to calculate the function $T^3$ when you say it is a "hefty task"?
$endgroup$
– Anthony Ter
Jan 29 at 6:37
$begingroup$
@AnthonyTer I am first calculating the image of standard basis of $R^3$ under $T^3$, then form the matrix of transformation of $T^3$ and from there I would calculate the rank of this matrix and use dimension theorem
$endgroup$
– Abhay
Jan 29 at 6:38
$begingroup$
Is there any easier way to do this?
$endgroup$
– Abhay
Jan 29 at 6:48
$begingroup$
Just form the matrix of transformation $T$ first then cube it, find nullity of that matrix.
$endgroup$
– Anthony Ter
Jan 29 at 6:48
add a comment |
$begingroup$
$T:R^3to R^3$
$T(x,y,z)=(x+3y+2z,3x+4y+z,2x+y-z)$
I want to find the dimension of the Null Space of $T^3$
Here Nullity of $T$ is not $0$, so I can't say anything directly about the nullity of $T^3$
Normally for the nullity of T, I would form the matrix of transformation by take the image of the standard basis of $R^3$, then row reduce to calculate the rank and use the dimension theorem.
But doing the same for $T^3$ is turning out to be hefty task and I am looking for a easier way.
linear-algebra vector-spaces linear-transformations
asked Jan 29 at 6:32
AbhayAbhay
3789
$endgroup$
$T:R^3to R^3$
$T(x,y,z)=(x+3y+2z,3x+4y+z,2x+y-z)$
I want to find the dimension of the Null Space of $T^3$
Here Nullity of $T$ is not $0$, so I can't say anything directly about the nullity of $T^3$
Normally for the nullity of T, I would form the matrix of transformation by take the image of the standard basis of $R^3$, then row reduce to calculate the rank and use the dimension theorem.
But doing the same for $T^3$ is turning out to be hefty task and I am looking for a easier way.
linear-algebra vector-spaces linear-transformations
linear-algebra vector-spaces linear-transformations
asked Jan 29 at 6:32
AbhayAbhay
3789
asked Jan 29 at 6:32
AbhayAbhay
3789
asked Jan 29 at 6:32
AbhayAbhay
3789
asked Jan 29 at 6:32
AbhayAbhay
3789
asked Jan 29 at 6:32
AbhayAbhay
3789
3789
$begingroup$
Are you calculating the matrix $T^3$ or trying to calculate the function $T^3$ when you say it is a "hefty task"?
$endgroup$
– Anthony Ter
Jan 29 at 6:37
$begingroup$
@AnthonyTer I am first calculating the image of standard basis of $R^3$ under $T^3$, then form the matrix of transformation of $T^3$ and from there I would calculate the rank of this matrix and use dimension theorem
$endgroup$
– Abhay
Jan 29 at 6:38
$begingroup$
Is there any easier way to do this?
$endgroup$
– Abhay
Jan 29 at 6:48
$begingroup$
Just form the matrix of transformation $T$ first then cube it, find nullity of that matrix.
$endgroup$
– Anthony Ter
Jan 29 at 6:48
add a comment |
$begingroup$
Are you calculating the matrix $T^3$ or trying to calculate the function $T^3$ when you say it is a "hefty task"?
$endgroup$
– Anthony Ter
Jan 29 at 6:37
$begingroup$
@AnthonyTer I am first calculating the image of standard basis of $R^3$ under $T^3$, then form the matrix of transformation of $T^3$ and from there I would calculate the rank of this matrix and use dimension theorem
$endgroup$
– Abhay
Jan 29 at 6:38
$begingroup$
Is there any easier way to do this?
$endgroup$
– Abhay
Jan 29 at 6:48
$begingroup$
Just form the matrix of transformation $T$ first then cube it, find nullity of that matrix.
$endgroup$
– Anthony Ter
Jan 29 at 6:48
$begingroup$
Are you calculating the matrix $T^3$ or trying to calculate the function $T^3$ when you say it is a "hefty task"?
$endgroup$
– Anthony Ter
Jan 29 at 6:37
$begingroup$
Are you calculating the matrix $T^3$ or trying to calculate the function $T^3$ when you say it is a "hefty task"?
$endgroup$
– Anthony Ter
Jan 29 at 6:37
$begingroup$
@AnthonyTer I am first calculating the image of standard basis of $R^3$ under $T^3$, then form the matrix of transformation of $T^3$ and from there I would calculate the rank of this matrix and use dimension theorem
$endgroup$
– Abhay
Jan 29 at 6:38
$begingroup$
@AnthonyTer I am first calculating the image of standard basis of $R^3$ under $T^3$, then form the matrix of transformation of $T^3$ and from there I would calculate the rank of this matrix and use dimension theorem
$endgroup$
– Abhay
Jan 29 at 6:38
$begingroup$
Is there any easier way to do this?
$endgroup$
– Abhay
Jan 29 at 6:48
$begingroup$
Is there any easier way to do this?
$endgroup$
– Abhay
Jan 29 at 6:48
$begingroup$
Just form the matrix of transformation $T$ first then cube it, find nullity of that matrix.
$endgroup$
– Anthony Ter
Jan 29 at 6:48
$begingroup$
Just form the matrix of transformation $T$ first then cube it, find nullity of that matrix.
$endgroup$
– Anthony Ter
Jan 29 at 6:48
add a comment |
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$begingroup$
Are you calculating the matrix $T^3$ or trying to calculate the function $T^3$ when you say it is a "hefty task"?
$endgroup$
– Anthony Ter
Jan 29 at 6:37
$begingroup$
@AnthonyTer I am first calculating the image of standard basis of $R^3$ under $T^3$, then form the matrix of transformation of $T^3$ and from there I would calculate the rank of this matrix and use dimension theorem
$endgroup$
– Abhay
Jan 29 at 6:38
$begingroup$
Is there any easier way to do this?
$endgroup$
– Abhay
Jan 29 at 6:48
$begingroup$
Just form the matrix of transformation $T$ first then cube it, find nullity of that matrix.
$endgroup$
– Anthony Ter
Jan 29 at 6:48