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What does this linear transformation do?
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The following is given:
- $F_4$ is the vector space of all symmetrical matrices
- $F = operatornameSpanleftbeginbmatrix 1 & 0 \ 0 & 0 endbmatrix; beginbmatrix 0 & 1 \ 1 & 0 endbmatrix ; beginbmatrix 0 & 0 \ 0 & 1 endbmatrix right$
- $ L:F rightarrow F : A rightarrow Abeginbmatrix 0 & 0 \ 0 & 1 endbmatrix + beginbmatrix 0 & 0 \ 0 & 1 endbmatrixA$
The following is asked:
- What does this linear transformation do ?
- Give an exact description of the kernel and range of the transformation.
- Give the matrix representation M of this transformation
- Eigenvalues and Eigenvectors of the transformation
- What is the dimension of the Eigenspace ?
The kernel, range, matrix representation, eigenvalues and eigenvectors are pretty straightforward. I am struggling with the explanation of what this transformation does and the dimension of the Eigenspace. Can someone explain to me what this transformation does and hint me on the eigenspace question ?
linear-algebra linear-transformations
edited Jul 30 at 1:04
Michael Hardy
204k23185461
asked Jul 30 at 1:01
mbouchi
112
add a comment |Â
up vote
2
down vote
favorite
The following is given:
- $F_4$ is the vector space of all symmetrical matrices
- $F = operatornameSpanleftbeginbmatrix 1 & 0 \ 0 & 0 endbmatrix; beginbmatrix 0 & 1 \ 1 & 0 endbmatrix ; beginbmatrix 0 & 0 \ 0 & 1 endbmatrix right$
- $ L:F rightarrow F : A rightarrow Abeginbmatrix 0 & 0 \ 0 & 1 endbmatrix + beginbmatrix 0 & 0 \ 0 & 1 endbmatrixA$
The following is asked:
- What does this linear transformation do ?
- Give an exact description of the kernel and range of the transformation.
- Give the matrix representation M of this transformation
- Eigenvalues and Eigenvectors of the transformation
- What is the dimension of the Eigenspace ?
The kernel, range, matrix representation, eigenvalues and eigenvectors are pretty straightforward. I am struggling with the explanation of what this transformation does and the dimension of the Eigenspace. Can someone explain to me what this transformation does and hint me on the eigenspace question ?
linear-algebra linear-transformations
edited Jul 30 at 1:04
Michael Hardy
204k23185461
asked Jul 30 at 1:01
mbouchi
112
Have you written down the 5th point correctly? There are three eigenspaces ...
– ancientmathematician
Jul 30 at 8:59
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
The following is given:
- $F_4$ is the vector space of all symmetrical matrices
- $F = operatornameSpanleftbeginbmatrix 1 & 0 \ 0 & 0 endbmatrix; beginbmatrix 0 & 1 \ 1 & 0 endbmatrix ; beginbmatrix 0 & 0 \ 0 & 1 endbmatrix right$
- $ L:F rightarrow F : A rightarrow Abeginbmatrix 0 & 0 \ 0 & 1 endbmatrix + beginbmatrix 0 & 0 \ 0 & 1 endbmatrixA$
The following is asked:
- What does this linear transformation do ?
- Give an exact description of the kernel and range of the transformation.
- Give the matrix representation M of this transformation
- Eigenvalues and Eigenvectors of the transformation
- What is the dimension of the Eigenspace ?
The kernel, range, matrix representation, eigenvalues and eigenvectors are pretty straightforward. I am struggling with the explanation of what this transformation does and the dimension of the Eigenspace. Can someone explain to me what this transformation does and hint me on the eigenspace question ?
linear-algebra linear-transformations
edited Jul 30 at 1:04
Michael Hardy
204k23185461
asked Jul 30 at 1:01
mbouchi
112
The following is given:
- $F_4$ is the vector space of all symmetrical matrices
- $F = operatornameSpanleftbeginbmatrix 1 & 0 \ 0 & 0 endbmatrix; beginbmatrix 0 & 1 \ 1 & 0 endbmatrix ; beginbmatrix 0 & 0 \ 0 & 1 endbmatrix right$
- $ L:F rightarrow F : A rightarrow Abeginbmatrix 0 & 0 \ 0 & 1 endbmatrix + beginbmatrix 0 & 0 \ 0 & 1 endbmatrixA$
The following is asked:
- What does this linear transformation do ?
- Give an exact description of the kernel and range of the transformation.
- Give the matrix representation M of this transformation
- Eigenvalues and Eigenvectors of the transformation
- What is the dimension of the Eigenspace ?
The kernel, range, matrix representation, eigenvalues and eigenvectors are pretty straightforward. I am struggling with the explanation of what this transformation does and the dimension of the Eigenspace. Can someone explain to me what this transformation does and hint me on the eigenspace question ?
linear-algebra linear-transformations
edited Jul 30 at 1:04
Michael Hardy
204k23185461
asked Jul 30 at 1:01
mbouchi
112
edited Jul 30 at 1:04
Michael Hardy
204k23185461
edited Jul 30 at 1:04
Michael Hardy
204k23185461
edited Jul 30 at 1:04
Michael Hardy
204k23185461
204k23185461
asked Jul 30 at 1:01
mbouchi
112
asked Jul 30 at 1:01
mbouchi
112
asked Jul 30 at 1:01
mbouchi
112
112
Have you written down the 5th point correctly? There are three eigenspaces ...
– ancientmathematician
Jul 30 at 8:59
add a comment |Â
Have you written down the 5th point correctly? There are three eigenspaces ...
– ancientmathematician
Jul 30 at 8:59
Have you written down the 5th point correctly? There are three eigenspaces ...
– ancientmathematician
Jul 30 at 8:59
Have you written down the 5th point correctly? There are three eigenspaces ...
– ancientmathematician
Jul 30 at 8:59
add a comment |Â
1 Answer
1
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Hint Write $A= beginbmatrix a & b \ c &d endbmatrix$. Then
$$L(A)= beginbmatrix a & b \ c &d endbmatrixrightarrow Abeginbmatrix 0 & 0 \ 0 & 1 endbmatrix + beginbmatrix 0 & 0 \ 0 & 1 endbmatrixbeginbmatrix a & b \ c &d endbmatrix= beginbmatrix ?? & ?? \ ?? &??endbmatrix$$
-Kernel: Solve $L(A)=0$.
-Range: describe all values of $L(A)$.
-Matrix Representation: calculate $L$ of the cannonical basis.
-Eigenvalues/Eigenvectors: For which $A$ do you get $L(A)=lambda A$?
answered Jul 30 at 1:12
N. S.
97.7k5105197
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Hint Write $A= beginbmatrix a & b \ c &d endbmatrix$. Then
$$L(A)= beginbmatrix a & b \ c &d endbmatrixrightarrow Abeginbmatrix 0 & 0 \ 0 & 1 endbmatrix + beginbmatrix 0 & 0 \ 0 & 1 endbmatrixbeginbmatrix a & b \ c &d endbmatrix= beginbmatrix ?? & ?? \ ?? &??endbmatrix$$
-Kernel: Solve $L(A)=0$.
-Range: describe all values of $L(A)$.
-Matrix Representation: calculate $L$ of the cannonical basis.
-Eigenvalues/Eigenvectors: For which $A$ do you get $L(A)=lambda A$?
answered Jul 30 at 1:12
N. S.
97.7k5105197
add a comment |Â
up vote
1
down vote
Hint Write $A= beginbmatrix a & b \ c &d endbmatrix$. Then
$$L(A)= beginbmatrix a & b \ c &d endbmatrixrightarrow Abeginbmatrix 0 & 0 \ 0 & 1 endbmatrix + beginbmatrix 0 & 0 \ 0 & 1 endbmatrixbeginbmatrix a & b \ c &d endbmatrix= beginbmatrix ?? & ?? \ ?? &??endbmatrix$$
-Kernel: Solve $L(A)=0$.
-Range: describe all values of $L(A)$.
-Matrix Representation: calculate $L$ of the cannonical basis.
-Eigenvalues/Eigenvectors: For which $A$ do you get $L(A)=lambda A$?
answered Jul 30 at 1:12
N. S.
97.7k5105197
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Hint Write $A= beginbmatrix a & b \ c &d endbmatrix$. Then
$$L(A)= beginbmatrix a & b \ c &d endbmatrixrightarrow Abeginbmatrix 0 & 0 \ 0 & 1 endbmatrix + beginbmatrix 0 & 0 \ 0 & 1 endbmatrixbeginbmatrix a & b \ c &d endbmatrix= beginbmatrix ?? & ?? \ ?? &??endbmatrix$$
-Kernel: Solve $L(A)=0$.
-Range: describe all values of $L(A)$.
-Matrix Representation: calculate $L$ of the cannonical basis.
-Eigenvalues/Eigenvectors: For which $A$ do you get $L(A)=lambda A$?
answered Jul 30 at 1:12
N. S.
97.7k5105197
Hint Write $A= beginbmatrix a & b \ c &d endbmatrix$. Then
$$L(A)= beginbmatrix a & b \ c &d endbmatrixrightarrow Abeginbmatrix 0 & 0 \ 0 & 1 endbmatrix + beginbmatrix 0 & 0 \ 0 & 1 endbmatrixbeginbmatrix a & b \ c &d endbmatrix= beginbmatrix ?? & ?? \ ?? &??endbmatrix$$
-Kernel: Solve $L(A)=0$.
-Range: describe all values of $L(A)$.
-Matrix Representation: calculate $L$ of the cannonical basis.
-Eigenvalues/Eigenvectors: For which $A$ do you get $L(A)=lambda A$?
answered Jul 30 at 1:12
N. S.
97.7k5105197
answered Jul 30 at 1:12
N. S.
97.7k5105197
answered Jul 30 at 1:12
N. S.
97.7k5105197
answered Jul 30 at 1:12
N. S.
97.7k5105197
97.7k5105197
add a comment |Â
add a comment |Â
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Have you written down the 5th point correctly? There are three eigenspaces ...
– ancientmathematician
Jul 30 at 8:59