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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.

3

1

$begingroup$

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

share|cite|improve this question

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 | 

3

1

$begingroup$

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

share|cite|improve this question

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

share|cite|improve this question

asked Mar 19 at 23:28

crt0476crt0476

161

$endgroup$

share|cite|improve this question

asked Mar 19 at 23:28

crt0476crt0476

161

share|cite|improve this question

asked Mar 19 at 23:28

crt0476crt0476

161

asked Mar 19 at 23:28

crt0476crt0476

161

asked Mar 19 at 23:28

crt0476crt0476

161

1 Answer
1

active

oldest

votes

2

$begingroup$

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 $Lambda^r(operatorname{id_{U}}) = operatorname{id}|_{Lambda^r(U)}$).

share|cite|improve this answer

answered Mar 20 at 10:12

levaplevap

48k33274

$endgroup$

add a comment | 

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2

$begingroup$

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 $Lambda^r(operatorname{id_{U}}) = operatorname{id}|_{Lambda^r(U)}$).

share|cite|improve this answer

answered Mar 20 at 10:12

levaplevap

48k33274

$endgroup$

add a comment | 

2

$begingroup$

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 $Lambda^r(operatorname{id_{U}}) = operatorname{id}|_{Lambda^r(U)}$).

share|cite|improve this answer

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 $Lambda^r(operatorname{id_{U}}) = operatorname{id}|_{Lambda^r(U)}$).

share|cite|improve this answer

answered Mar 20 at 10:12

levaplevap

48k33274

$endgroup$

share|cite|improve this answer

answered Mar 20 at 10:12

levaplevap

48k33274

answered Mar 20 at 10:12

levaplevap

48k33274

answered Mar 20 at 10:12

levaplevap

48k33274