Mixing
A question about vector space, sub-spaces and linear transformation
$begingroup$
I have a homework question that I don't know how to start:
let $V$ be a vector space under field $F$, $dim(V) = n$.
let $T: V to V$ be a linear transformation.
prove if $U subseteq V$, and $dim(U) = k$, such that for every $u in U$, $T(u) in U$, then there exists a basis $B$ of $V$ such that:
$[T]^B_B =$
$
begin{bmatrix}
A_1 & A_3 \
0 & A_2 \
end{bmatrix}
$
So that $A_3 ∈ M_{k×(n−k)}(F)$, $A_2 in M_{(n−k)×(n−k)}(F)$ and $A_1 in M_{k×k}(F)$
Thank you for your help!
linear-algebra matrices vector-spaces linear-transformations
edited Jan 2 at 21:33
A.Γ.
22.7k32656
asked Jan 2 at 20:09
assafassaf
133
$endgroup$
add a comment |
$begingroup$
I have a homework question that I don't know how to start:
let $V$ be a vector space under field $F$, $dim(V) = n$.
let $T: V to V$ be a linear transformation.
prove if $U subseteq V$, and $dim(U) = k$, such that for every $u in U$, $T(u) in U$, then there exists a basis $B$ of $V$ such that:
$[T]^B_B =$
$
begin{bmatrix}
A_1 & A_3 \
0 & A_2 \
end{bmatrix}
$
So that $A_3 ∈ M_{k×(n−k)}(F)$, $A_2 in M_{(n−k)×(n−k)}(F)$ and $A_1 in M_{k×k}(F)$
Thank you for your help!
linear-algebra matrices vector-spaces linear-transformations
edited Jan 2 at 21:33
A.Γ.
22.7k32656
asked Jan 2 at 20:09
assafassaf
133
$endgroup$
1
$begingroup$
Please use MathJax to format your post so it's easier to read.
$endgroup$
– saulspatz
Jan 2 at 20:17
1
$begingroup$
I edited it, it just took me some time.
$endgroup$
– assaf
Jan 2 at 20:27
add a comment |
$begingroup$
I have a homework question that I don't know how to start:
let $V$ be a vector space under field $F$, $dim(V) = n$.
let $T: V to V$ be a linear transformation.
prove if $U subseteq V$, and $dim(U) = k$, such that for every $u in U$, $T(u) in U$, then there exists a basis $B$ of $V$ such that:
$[T]^B_B =$
$
begin{bmatrix}
A_1 & A_3 \
0 & A_2 \
end{bmatrix}
$
So that $A_3 ∈ M_{k×(n−k)}(F)$, $A_2 in M_{(n−k)×(n−k)}(F)$ and $A_1 in M_{k×k}(F)$
Thank you for your help!
linear-algebra matrices vector-spaces linear-transformations
edited Jan 2 at 21:33
A.Γ.
22.7k32656
asked Jan 2 at 20:09
assafassaf
133
$endgroup$
I have a homework question that I don't know how to start:
let $V$ be a vector space under field $F$, $dim(V) = n$.
let $T: V to V$ be a linear transformation.
prove if $U subseteq V$, and $dim(U) = k$, such that for every $u in U$, $T(u) in U$, then there exists a basis $B$ of $V$ such that:
$[T]^B_B =$
$
begin{bmatrix}
A_1 & A_3 \
0 & A_2 \
end{bmatrix}
$
So that $A_3 ∈ M_{k×(n−k)}(F)$, $A_2 in M_{(n−k)×(n−k)}(F)$ and $A_1 in M_{k×k}(F)$
Thank you for your help!
linear-algebra matrices vector-spaces linear-transformations
linear-algebra matrices vector-spaces linear-transformations
edited Jan 2 at 21:33
A.Γ.
22.7k32656
asked Jan 2 at 20:09
assafassaf
133
edited Jan 2 at 21:33
A.Γ.
22.7k32656
asked Jan 2 at 20:09
assafassaf
133
edited Jan 2 at 21:33
A.Γ.
22.7k32656
edited Jan 2 at 21:33
A.Γ.
22.7k32656
edited Jan 2 at 21:33
A.Γ.
22.7k32656
22.7k32656
asked Jan 2 at 20:09
assafassaf
133
asked Jan 2 at 20:09
assafassaf
133
asked Jan 2 at 20:09
assafassaf
133
133
1
$begingroup$
Please use MathJax to format your post so it's easier to read.
$endgroup$
– saulspatz
Jan 2 at 20:17
1
$begingroup$
I edited it, it just took me some time.
$endgroup$
– assaf
Jan 2 at 20:27
add a comment |
1
$begingroup$
Please use MathJax to format your post so it's easier to read.
$endgroup$
– saulspatz
Jan 2 at 20:17
1
$begingroup$
I edited it, it just took me some time.
$endgroup$
– assaf
Jan 2 at 20:27
1
1
$begingroup$
Please use MathJax to format your post so it's easier to read.
$endgroup$
– saulspatz
Jan 2 at 20:17
$begingroup$
Please use MathJax to format your post so it's easier to read.
$endgroup$
– saulspatz
Jan 2 at 20:17
1
1
$begingroup$
I edited it, it just took me some time.
$endgroup$
– assaf
Jan 2 at 20:27
$begingroup$
I edited it, it just took me some time.
$endgroup$
– assaf
Jan 2 at 20:27
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
Hint: Take a basis $u_1,dots,u_k$ of $U$ and complete it to a basis $u_1,dots,u_k,u_{k+1},dots,u_n$ of $V$.
answered Jan 2 at 21:31
ScientificaScientifica
6,37641335
$endgroup$
add a comment |
$begingroup$
Hint: try to construct the basis $B$. Start with a basis of $U$ ($k$ vectors) and complete it to a basis in $V$ by $n-k$ linearly independent vectors. How does the matrix of $T$ look like in this basis?
answered Jan 2 at 21:31
A.Γ.A.Γ.
22.7k32656
$endgroup$
add a comment |
$begingroup$
First you argue that there is a basis of $V$ such that any element of $U$ can be written as $(u_1,dots,u_k,0,dots,0)in V$.
Using this and that by assumption $T(U)subset U$ we deduce that the matrix representation of $T$ in this basis will be:
$$
begin{bmatrix}
A & B \
C & D \
end{bmatrix}
$$
where $A$ is a $ktimes k$ matrix and $C=0$.
answered Jan 2 at 21:46
Test123Test123
2,762828
$endgroup$
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint: Take a basis $u_1,dots,u_k$ of $U$ and complete it to a basis $u_1,dots,u_k,u_{k+1},dots,u_n$ of $V$.
answered Jan 2 at 21:31
ScientificaScientifica
6,37641335
$endgroup$
add a comment |
$begingroup$
Hint: Take a basis $u_1,dots,u_k$ of $U$ and complete it to a basis $u_1,dots,u_k,u_{k+1},dots,u_n$ of $V$.
answered Jan 2 at 21:31
ScientificaScientifica
6,37641335
$endgroup$
add a comment |
$begingroup$
Hint: Take a basis $u_1,dots,u_k$ of $U$ and complete it to a basis $u_1,dots,u_k,u_{k+1},dots,u_n$ of $V$.
answered Jan 2 at 21:31
ScientificaScientifica
6,37641335
$endgroup$
Hint: Take a basis $u_1,dots,u_k$ of $U$ and complete it to a basis $u_1,dots,u_k,u_{k+1},dots,u_n$ of $V$.
answered Jan 2 at 21:31
ScientificaScientifica
6,37641335
answered Jan 2 at 21:31
ScientificaScientifica
6,37641335
answered Jan 2 at 21:31
ScientificaScientifica
6,37641335
answered Jan 2 at 21:31
ScientificaScientifica
6,37641335
6,37641335
add a comment |
add a comment |
$begingroup$
Hint: try to construct the basis $B$. Start with a basis of $U$ ($k$ vectors) and complete it to a basis in $V$ by $n-k$ linearly independent vectors. How does the matrix of $T$ look like in this basis?
answered Jan 2 at 21:31
A.Γ.A.Γ.
22.7k32656
$endgroup$
add a comment |
$begingroup$
Hint: try to construct the basis $B$. Start with a basis of $U$ ($k$ vectors) and complete it to a basis in $V$ by $n-k$ linearly independent vectors. How does the matrix of $T$ look like in this basis?
answered Jan 2 at 21:31
A.Γ.A.Γ.
22.7k32656
$endgroup$
add a comment |
$begingroup$
Hint: try to construct the basis $B$. Start with a basis of $U$ ($k$ vectors) and complete it to a basis in $V$ by $n-k$ linearly independent vectors. How does the matrix of $T$ look like in this basis?
answered Jan 2 at 21:31
A.Γ.A.Γ.
22.7k32656
$endgroup$
Hint: try to construct the basis $B$. Start with a basis of $U$ ($k$ vectors) and complete it to a basis in $V$ by $n-k$ linearly independent vectors. How does the matrix of $T$ look like in this basis?
answered Jan 2 at 21:31
A.Γ.A.Γ.
22.7k32656
answered Jan 2 at 21:31
A.Γ.A.Γ.
22.7k32656
answered Jan 2 at 21:31
A.Γ.A.Γ.
22.7k32656
answered Jan 2 at 21:31
A.Γ.A.Γ.
22.7k32656
22.7k32656
add a comment |
add a comment |
$begingroup$
First you argue that there is a basis of $V$ such that any element of $U$ can be written as $(u_1,dots,u_k,0,dots,0)in V$.
Using this and that by assumption $T(U)subset U$ we deduce that the matrix representation of $T$ in this basis will be:
$$
begin{bmatrix}
A & B \
C & D \
end{bmatrix}
$$
where $A$ is a $ktimes k$ matrix and $C=0$.
answered Jan 2 at 21:46
Test123Test123
2,762828
$endgroup$
add a comment |
$begingroup$
First you argue that there is a basis of $V$ such that any element of $U$ can be written as $(u_1,dots,u_k,0,dots,0)in V$.
Using this and that by assumption $T(U)subset U$ we deduce that the matrix representation of $T$ in this basis will be:
$$
begin{bmatrix}
A & B \
C & D \
end{bmatrix}
$$
where $A$ is a $ktimes k$ matrix and $C=0$.
answered Jan 2 at 21:46
Test123Test123
2,762828
$endgroup$
add a comment |
$begingroup$
First you argue that there is a basis of $V$ such that any element of $U$ can be written as $(u_1,dots,u_k,0,dots,0)in V$.
Using this and that by assumption $T(U)subset U$ we deduce that the matrix representation of $T$ in this basis will be:
$$
begin{bmatrix}
A & B \
C & D \
end{bmatrix}
$$
where $A$ is a $ktimes k$ matrix and $C=0$.
answered Jan 2 at 21:46
Test123Test123
2,762828
$endgroup$
First you argue that there is a basis of $V$ such that any element of $U$ can be written as $(u_1,dots,u_k,0,dots,0)in V$.
Using this and that by assumption $T(U)subset U$ we deduce that the matrix representation of $T$ in this basis will be:
$$
begin{bmatrix}
A & B \
C & D \
end{bmatrix}
$$
where $A$ is a $ktimes k$ matrix and $C=0$.
answered Jan 2 at 21:46
Test123Test123
2,762828
answered Jan 2 at 21:46
Test123Test123
2,762828
answered Jan 2 at 21:46
Test123Test123
2,762828
answered Jan 2 at 21:46
Test123Test123
2,762828
2,762828
add a comment |
add a comment |
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1
$begingroup$
Please use MathJax to format your post so it's easier to read.
$endgroup$
– saulspatz
Jan 2 at 20:17
1
$begingroup$
I edited it, it just took me some time.
$endgroup$
– assaf
Jan 2 at 20:27