Mixing
Different representation matrices for the same linear transformation?
Clash Royale CLAN TAG#URR8PPP
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0
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My question is about linear transformations and their representation matrices.
Given the matrices $A, B,$ Such that:
$$A=
beginbmatrix
1 & 0 & 0 \
0 & 2 & 0 \
0 & 0 & 3
endbmatrix,
B=
beginbmatrix
2 & 0 & 0 \
0 & 1 & 0 \
0 & 0 & 3
endbmatrix$$
Is there a linear transformation $T:R^3 to R^3$ such that both $A$ and $B$ are the representation matrix of $T?$
I heva no idea how to slve that problem, maybe the answer is yes because the both matrices have the same eigen values?
Is there any other way to get the solution?
Thanks for help!!
linear-algebra matrices eigenvalues-eigenvectors linear-transformations
asked Jul 26 at 7:38
D.Rotnemer
11216
add a comment |Â
up vote
0
down vote
favorite
My question is about linear transformations and their representation matrices.
Given the matrices $A, B,$ Such that:
$$A=
beginbmatrix
1 & 0 & 0 \
0 & 2 & 0 \
0 & 0 & 3
endbmatrix,
B=
beginbmatrix
2 & 0 & 0 \
0 & 1 & 0 \
0 & 0 & 3
endbmatrix$$
Is there a linear transformation $T:R^3 to R^3$ such that both $A$ and $B$ are the representation matrix of $T?$
I heva no idea how to slve that problem, maybe the answer is yes because the both matrices have the same eigen values?
Is there any other way to get the solution?
Thanks for help!!
linear-algebra matrices eigenvalues-eigenvectors linear-transformations
asked Jul 26 at 7:38
D.Rotnemer
11216
1
What is your choice of basis?
– ITA
Jul 26 at 7:42
There is no basis specified in the question, I think the standard basis.
– D.Rotnemer
Jul 26 at 7:45
1
I think Fred's answer below is it. The matrix representation always depends on a choice of basis for you domain and co-domain. Without specifying what they are exactly, the answer would also depend.
– ITA
Jul 26 at 7:55
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
My question is about linear transformations and their representation matrices.
Given the matrices $A, B,$ Such that:
$$A=
beginbmatrix
1 & 0 & 0 \
0 & 2 & 0 \
0 & 0 & 3
endbmatrix,
B=
beginbmatrix
2 & 0 & 0 \
0 & 1 & 0 \
0 & 0 & 3
endbmatrix$$
Is there a linear transformation $T:R^3 to R^3$ such that both $A$ and $B$ are the representation matrix of $T?$
I heva no idea how to slve that problem, maybe the answer is yes because the both matrices have the same eigen values?
Is there any other way to get the solution?
Thanks for help!!
linear-algebra matrices eigenvalues-eigenvectors linear-transformations
asked Jul 26 at 7:38
D.Rotnemer
11216
My question is about linear transformations and their representation matrices.
Given the matrices $A, B,$ Such that:
$$A=
beginbmatrix
1 & 0 & 0 \
0 & 2 & 0 \
0 & 0 & 3
endbmatrix,
B=
beginbmatrix
2 & 0 & 0 \
0 & 1 & 0 \
0 & 0 & 3
endbmatrix$$
Is there a linear transformation $T:R^3 to R^3$ such that both $A$ and $B$ are the representation matrix of $T?$
I heva no idea how to slve that problem, maybe the answer is yes because the both matrices have the same eigen values?
Is there any other way to get the solution?
Thanks for help!!
linear-algebra matrices eigenvalues-eigenvectors linear-transformations
asked Jul 26 at 7:38
D.Rotnemer
11216
asked Jul 26 at 7:38
D.Rotnemer
11216
asked Jul 26 at 7:38
D.Rotnemer
11216
asked Jul 26 at 7:38
D.Rotnemer
11216
11216
1
What is your choice of basis?
– ITA
Jul 26 at 7:42
There is no basis specified in the question, I think the standard basis.
– D.Rotnemer
Jul 26 at 7:45
1
I think Fred's answer below is it. The matrix representation always depends on a choice of basis for you domain and co-domain. Without specifying what they are exactly, the answer would also depend.
– ITA
Jul 26 at 7:55
add a comment |Â
1
What is your choice of basis?
– ITA
Jul 26 at 7:42
There is no basis specified in the question, I think the standard basis.
– D.Rotnemer
Jul 26 at 7:45
1
I think Fred's answer below is it. The matrix representation always depends on a choice of basis for you domain and co-domain. Without specifying what they are exactly, the answer would also depend.
– ITA
Jul 26 at 7:55
1
1
What is your choice of basis?
– ITA
Jul 26 at 7:42
What is your choice of basis?
– ITA
Jul 26 at 7:42
There is no basis specified in the question, I think the standard basis.
– D.Rotnemer
Jul 26 at 7:45
There is no basis specified in the question, I think the standard basis.
– D.Rotnemer
Jul 26 at 7:45
1
1
I think Fred's answer below is it. The matrix representation always depends on a choice of basis for you domain and co-domain. Without specifying what they are exactly, the answer would also depend.
– ITA
Jul 26 at 7:55
I think Fred's answer below is it. The matrix representation always depends on a choice of basis for you domain and co-domain. Without specifying what they are exactly, the answer would also depend.
– ITA
Jul 26 at 7:55
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
1
down vote
accepted
Let $B_1:=b_1,b_2,b_3$ be a basis of $V:=mathbb R^3$ and define the linear mapping $T:V to V$ by
$T(b_1)=b_1, T(b_2)=2b_2$ and $T(b_3)=3 b_3$.
Then $T$ has the representation matrix $A$.
If $B_2:=b_2,b_1,b_3$, then the representation matrix of $T$ is $B$
answered Jul 26 at 7:48
Fred
37.2k1237
add a comment |Â
up vote
1
down vote
Yes the two matrices are similar and they represent the same transformation indeed
$$B=P^-1AP$$
$$ beginbmatrix
2 & 0 & 0 \
0 & 1 & 0 \
0 & 0 & 3
endbmatrix= beginbmatrix
0 & 1 & 0 \
1 & 0 & 0 \
0 & 0 & 1
endbmatrix beginbmatrix
1 & 0 & 0 \
0 & 2 & 0 \
0 & 0 & 3
endbmatrix beginbmatrix
0 & 1 & 0 \
1 & 0 & 0 \
0 & 0 & 1
endbmatrix$$
answered Jul 26 at 7:58
gimusi
65k73583
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Let $B_1:=b_1,b_2,b_3$ be a basis of $V:=mathbb R^3$ and define the linear mapping $T:V to V$ by
$T(b_1)=b_1, T(b_2)=2b_2$ and $T(b_3)=3 b_3$.
Then $T$ has the representation matrix $A$.
If $B_2:=b_2,b_1,b_3$, then the representation matrix of $T$ is $B$
answered Jul 26 at 7:48
Fred
37.2k1237
add a comment |Â
up vote
1
down vote
accepted
Let $B_1:=b_1,b_2,b_3$ be a basis of $V:=mathbb R^3$ and define the linear mapping $T:V to V$ by
$T(b_1)=b_1, T(b_2)=2b_2$ and $T(b_3)=3 b_3$.
Then $T$ has the representation matrix $A$.
If $B_2:=b_2,b_1,b_3$, then the representation matrix of $T$ is $B$
answered Jul 26 at 7:48
Fred
37.2k1237
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Let $B_1:=b_1,b_2,b_3$ be a basis of $V:=mathbb R^3$ and define the linear mapping $T:V to V$ by
$T(b_1)=b_1, T(b_2)=2b_2$ and $T(b_3)=3 b_3$.
Then $T$ has the representation matrix $A$.
If $B_2:=b_2,b_1,b_3$, then the representation matrix of $T$ is $B$
answered Jul 26 at 7:48
Fred
37.2k1237
Let $B_1:=b_1,b_2,b_3$ be a basis of $V:=mathbb R^3$ and define the linear mapping $T:V to V$ by
$T(b_1)=b_1, T(b_2)=2b_2$ and $T(b_3)=3 b_3$.
Then $T$ has the representation matrix $A$.
If $B_2:=b_2,b_1,b_3$, then the representation matrix of $T$ is $B$
answered Jul 26 at 7:48
Fred
37.2k1237
answered Jul 26 at 7:48
Fred
37.2k1237
answered Jul 26 at 7:48
Fred
37.2k1237
answered Jul 26 at 7:48
Fred
37.2k1237
37.2k1237
add a comment |Â
add a comment |Â
up vote
1
down vote
Yes the two matrices are similar and they represent the same transformation indeed
$$B=P^-1AP$$
$$ beginbmatrix
2 & 0 & 0 \
0 & 1 & 0 \
0 & 0 & 3
endbmatrix= beginbmatrix
0 & 1 & 0 \
1 & 0 & 0 \
0 & 0 & 1
endbmatrix beginbmatrix
1 & 0 & 0 \
0 & 2 & 0 \
0 & 0 & 3
endbmatrix beginbmatrix
0 & 1 & 0 \
1 & 0 & 0 \
0 & 0 & 1
endbmatrix$$
answered Jul 26 at 7:58
gimusi
65k73583
add a comment |Â
up vote
1
down vote
Yes the two matrices are similar and they represent the same transformation indeed
$$B=P^-1AP$$
$$ beginbmatrix
2 & 0 & 0 \
0 & 1 & 0 \
0 & 0 & 3
endbmatrix= beginbmatrix
0 & 1 & 0 \
1 & 0 & 0 \
0 & 0 & 1
endbmatrix beginbmatrix
1 & 0 & 0 \
0 & 2 & 0 \
0 & 0 & 3
endbmatrix beginbmatrix
0 & 1 & 0 \
1 & 0 & 0 \
0 & 0 & 1
endbmatrix$$
answered Jul 26 at 7:58
gimusi
65k73583
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Yes the two matrices are similar and they represent the same transformation indeed
$$B=P^-1AP$$
$$ beginbmatrix
2 & 0 & 0 \
0 & 1 & 0 \
0 & 0 & 3
endbmatrix= beginbmatrix
0 & 1 & 0 \
1 & 0 & 0 \
0 & 0 & 1
endbmatrix beginbmatrix
1 & 0 & 0 \
0 & 2 & 0 \
0 & 0 & 3
endbmatrix beginbmatrix
0 & 1 & 0 \
1 & 0 & 0 \
0 & 0 & 1
endbmatrix$$
answered Jul 26 at 7:58
gimusi
65k73583
Yes the two matrices are similar and they represent the same transformation indeed
$$B=P^-1AP$$
$$ beginbmatrix
2 & 0 & 0 \
0 & 1 & 0 \
0 & 0 & 3
endbmatrix= beginbmatrix
0 & 1 & 0 \
1 & 0 & 0 \
0 & 0 & 1
endbmatrix beginbmatrix
1 & 0 & 0 \
0 & 2 & 0 \
0 & 0 & 3
endbmatrix beginbmatrix
0 & 1 & 0 \
1 & 0 & 0 \
0 & 0 & 1
endbmatrix$$
answered Jul 26 at 7:58
gimusi
65k73583
answered Jul 26 at 7:58
gimusi
65k73583
answered Jul 26 at 7:58
gimusi
65k73583
answered Jul 26 at 7:58
gimusi
65k73583
65k73583
add a comment |Â
add a comment |Â
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1
What is your choice of basis?
– ITA
Jul 26 at 7:42
There is no basis specified in the question, I think the standard basis.
– D.Rotnemer
Jul 26 at 7:45
1
I think Fred's answer below is it. The matrix representation always depends on a choice of basis for you domain and co-domain. Without specifying what they are exactly, the answer would also depend.
– ITA
Jul 26 at 7:55