The kernel of $bigwedge^r(T): bigwedge^r(W)to bigwedge^r(V)$?Determining the intersection of kernel and...
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The kernel of $bigwedge^r(T): bigwedge^r(W)to bigwedge^r(V)$?
Determining the intersection of kernel and image.Describe the image and kernel of a projectionHow to prove: Orthogonal complement of kernel = Row space?Dimension of Image/Kernel of Linear Transformationdirect sums, kernel and linear operator on a finite dimensional vector spaceExample of linear transformation on infinite dimensional vector spaceKernel and image of a matrix converting linear transformationBasis for Kernel and Trace of Square MatricesCan $Im(sigma)cap Ker(sigma)$ always be ${0}$?Find dimension of kernel.
$begingroup$
The full problem is:
Given $T: Wto V$, a linear transformation of $F$-vector spaces, such that $text{ker} T = 0$, show that the kernel of $bigwedge^r(T): bigwedge^r(W)to bigwedge^r(V)$ is also zero.
I'm confused on how to deal with this problem when $T$ is not an endomorphism. It would also be helpful to know what $bigwedge^r(T)$ is supposed to look like.
Thanks
linear-algebra linear-transformations exterior-algebra
asked Mar 19 at 23:28
crt0476crt0476
161
$endgroup$
add a comment |
$begingroup$
The full problem is:
Given $T: Wto V$, a linear transformation of $F$-vector spaces, such that $text{ker} T = 0$, show that the kernel of $bigwedge^r(T): bigwedge^r(W)to bigwedge^r(V)$ is also zero.
I'm confused on how to deal with this problem when $T$ is not an endomorphism. It would also be helpful to know what $bigwedge^r(T)$ is supposed to look like.
Thanks
linear-algebra linear-transformations exterior-algebra
asked Mar 19 at 23:28
crt0476crt0476
161
$endgroup$
1
$begingroup$
HINT: If $ker T = {0}$, then $T$ maps sets of $r$ linearly independent vectors to linearly independent vectors.
$endgroup$
– Ted Shifrin
Mar 19 at 23:32
add a comment |
$begingroup$
The full problem is:
Given $T: Wto V$, a linear transformation of $F$-vector spaces, such that $text{ker} T = 0$, show that the kernel of $bigwedge^r(T): bigwedge^r(W)to bigwedge^r(V)$ is also zero.
I'm confused on how to deal with this problem when $T$ is not an endomorphism. It would also be helpful to know what $bigwedge^r(T)$ is supposed to look like.
Thanks
linear-algebra linear-transformations exterior-algebra
asked Mar 19 at 23:28
crt0476crt0476
161
$endgroup$
The full problem is:
Given $T: Wto V$, a linear transformation of $F$-vector spaces, such that $text{ker} T = 0$, show that the kernel of $bigwedge^r(T): bigwedge^r(W)to bigwedge^r(V)$ is also zero.
I'm confused on how to deal with this problem when $T$ is not an endomorphism. It would also be helpful to know what $bigwedge^r(T)$ is supposed to look like.
Thanks
linear-algebra linear-transformations exterior-algebra
linear-algebra linear-transformations exterior-algebra
asked Mar 19 at 23:28
crt0476crt0476
161
asked Mar 19 at 23:28
crt0476crt0476
161
asked Mar 19 at 23:28
crt0476crt0476
161
asked Mar 19 at 23:28
crt0476crt0476
161
asked Mar 19 at 23:28
crt0476crt0476
161
161
1
$begingroup$
HINT: If $ker T = {0}$, then $T$ maps sets of $r$ linearly independent vectors to linearly independent vectors.
$endgroup$
– Ted Shifrin
Mar 19 at 23:32
add a comment |
1
$begingroup$
HINT: If $ker T = {0}$, then $T$ maps sets of $r$ linearly independent vectors to linearly independent vectors.
$endgroup$
– Ted Shifrin
Mar 19 at 23:32
1
1
$begingroup$
HINT: If $ker T = {0}$, then $T$ maps sets of $r$ linearly independent vectors to linearly independent vectors.
$endgroup$
– Ted Shifrin
Mar 19 at 23:32
$begingroup$
HINT: If $ker T = {0}$, then $T$ maps sets of $r$ linearly independent vectors to linearly independent vectors.
$endgroup$
– Ted Shifrin
Mar 19 at 23:32
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint: Use the fact that a linear transformation $T colon W rightarrow V$ is injective if and only if there exists a linear transformation $S colon V rightarrow W$ such that $S circ T = operatorname{id}_{W}$. Then use the functoriality of $Lambda$ (that is, $Lambda^r(S circ T) = Lambda^r(S) circ Lambda^r(T)$ and $Lambda^r(operatorname{id_{U}}) = operatorname{id}|_{Lambda^r(U)}$).
answered Mar 20 at 10:12
levaplevap
48k33274
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add a comment |
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$begingroup$
Hint: Use the fact that a linear transformation $T colon W rightarrow V$ is injective if and only if there exists a linear transformation $S colon V rightarrow W$ such that $S circ T = operatorname{id}_{W}$. Then use the functoriality of $Lambda$ (that is, $Lambda^r(S circ T) = Lambda^r(S) circ Lambda^r(T)$ and $Lambda^r(operatorname{id_{U}}) = operatorname{id}|_{Lambda^r(U)}$).
answered Mar 20 at 10:12
levaplevap
48k33274
$endgroup$
add a comment |
$begingroup$
Hint: Use the fact that a linear transformation $T colon W rightarrow V$ is injective if and only if there exists a linear transformation $S colon V rightarrow W$ such that $S circ T = operatorname{id}_{W}$. Then use the functoriality of $Lambda$ (that is, $Lambda^r(S circ T) = Lambda^r(S) circ Lambda^r(T)$ and $Lambda^r(operatorname{id_{U}}) = operatorname{id}|_{Lambda^r(U)}$).
answered Mar 20 at 10:12
levaplevap
48k33274
$endgroup$
add a comment |
$begingroup$
Hint: Use the fact that a linear transformation $T colon W rightarrow V$ is injective if and only if there exists a linear transformation $S colon V rightarrow W$ such that $S circ T = operatorname{id}_{W}$. Then use the functoriality of $Lambda$ (that is, $Lambda^r(S circ T) = Lambda^r(S) circ Lambda^r(T)$ and $Lambda^r(operatorname{id_{U}}) = operatorname{id}|_{Lambda^r(U)}$).
answered Mar 20 at 10:12
levaplevap
48k33274
$endgroup$
Hint: Use the fact that a linear transformation $T colon W rightarrow V$ is injective if and only if there exists a linear transformation $S colon V rightarrow W$ such that $S circ T = operatorname{id}_{W}$. Then use the functoriality of $Lambda$ (that is, $Lambda^r(S circ T) = Lambda^r(S) circ Lambda^r(T)$ and $Lambda^r(operatorname{id_{U}}) = operatorname{id}|_{Lambda^r(U)}$).
answered Mar 20 at 10:12
levaplevap
48k33274
answered Mar 20 at 10:12
levaplevap
48k33274
answered Mar 20 at 10:12
levaplevap
48k33274
answered Mar 20 at 10:12
levaplevap
48k33274
48k33274
add a comment |
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$begingroup$
HINT: If $ker T = {0}$, then $T$ maps sets of $r$ linearly independent vectors to linearly independent vectors.
$endgroup$
– Ted Shifrin
Mar 19 at 23:32