Range and Null Space are Complementary
I am trying to do the following question, which is Exercise 2 on page 241 of Linear Algebra by Hoffman and Kunze:
Let $T$ be a linear operator on the finite-dimensional space $V$, and let $R$ be the range [image] of $T$.
(a) Prove that $R$ has a complementary $T$-invariant subspace if and only if $R$ is independent of the null space $N$ of $T$ [$R cap N = 0$].
(b) If $R$ and $N$ are independent, prove that $N$ is the unique $T$-invariant subspace complementary to $R$.
I am having trouble with even starting this problem. I do not have any intuition as why this would be true.
linear-algebra abstract-algebra
asked Dec 3 '18 at 20:48
LinearGuy
1457
add a comment |
I am trying to do the following question, which is Exercise 2 on page 241 of Linear Algebra by Hoffman and Kunze:
Let $T$ be a linear operator on the finite-dimensional space $V$, and let $R$ be the range [image] of $T$.
(a) Prove that $R$ has a complementary $T$-invariant subspace if and only if $R$ is independent of the null space $N$ of $T$ [$R cap N = 0$].
(b) If $R$ and $N$ are independent, prove that $N$ is the unique $T$-invariant subspace complementary to $R$.
I am having trouble with even starting this problem. I do not have any intuition as why this would be true.
linear-algebra abstract-algebra
asked Dec 3 '18 at 20:48
LinearGuy
1457
1
You can use that the sum of the dimension of the range and of the null space is equal to the dimension of $V$. You then have two subspaces which by assumption do not intersect and whose sum of dimensions equals the dimension of the space.
– Eric
Dec 3 '18 at 21:24
By rank-nullity, $dim R + dim N = dim V$ and the question is asking about when we have a direct sum decomposition $V = R oplus N$.
– Trevor Gunn
Dec 3 '18 at 21:24
add a comment |
I am trying to do the following question, which is Exercise 2 on page 241 of Linear Algebra by Hoffman and Kunze:
Let $T$ be a linear operator on the finite-dimensional space $V$, and let $R$ be the range [image] of $T$.
(a) Prove that $R$ has a complementary $T$-invariant subspace if and only if $R$ is independent of the null space $N$ of $T$ [$R cap N = 0$].
(b) If $R$ and $N$ are independent, prove that $N$ is the unique $T$-invariant subspace complementary to $R$.
I am having trouble with even starting this problem. I do not have any intuition as why this would be true.
linear-algebra abstract-algebra
asked Dec 3 '18 at 20:48
LinearGuy
1457
I am trying to do the following question, which is Exercise 2 on page 241 of Linear Algebra by Hoffman and Kunze:
Let $T$ be a linear operator on the finite-dimensional space $V$, and let $R$ be the range [image] of $T$.
(a) Prove that $R$ has a complementary $T$-invariant subspace if and only if $R$ is independent of the null space $N$ of $T$ [$R cap N = 0$].
(b) If $R$ and $N$ are independent, prove that $N$ is the unique $T$-invariant subspace complementary to $R$.
I am having trouble with even starting this problem. I do not have any intuition as why this would be true.
linear-algebra abstract-algebra
linear-algebra abstract-algebra
asked Dec 3 '18 at 20:48
LinearGuy
1457
asked Dec 3 '18 at 20:48
LinearGuy
1457
edited Dec 3 '18 at 21:08
asked Dec 3 '18 at 20:48
LinearGuy
1457
asked Dec 3 '18 at 20:48
LinearGuy
1457
asked Dec 3 '18 at 20:48
LinearGuy
1457
1457
1
You can use that the sum of the dimension of the range and of the null space is equal to the dimension of $V$. You then have two subspaces which by assumption do not intersect and whose sum of dimensions equals the dimension of the space.
– Eric
Dec 3 '18 at 21:24
By rank-nullity, $dim R + dim N = dim V$ and the question is asking about when we have a direct sum decomposition $V = R oplus N$.
– Trevor Gunn
Dec 3 '18 at 21:24
add a comment |
1
You can use that the sum of the dimension of the range and of the null space is equal to the dimension of $V$. You then have two subspaces which by assumption do not intersect and whose sum of dimensions equals the dimension of the space.
– Eric
Dec 3 '18 at 21:24
By rank-nullity, $dim R + dim N = dim V$ and the question is asking about when we have a direct sum decomposition $V = R oplus N$.
– Trevor Gunn
Dec 3 '18 at 21:24
1
1
You can use that the sum of the dimension of the range and of the null space is equal to the dimension of $V$. You then have two subspaces which by assumption do not intersect and whose sum of dimensions equals the dimension of the space.
– Eric
Dec 3 '18 at 21:24
You can use that the sum of the dimension of the range and of the null space is equal to the dimension of $V$. You then have two subspaces which by assumption do not intersect and whose sum of dimensions equals the dimension of the space.
– Eric
Dec 3 '18 at 21:24
By rank-nullity, $dim R + dim N = dim V$ and the question is asking about when we have a direct sum decomposition $V = R oplus N$.
– Trevor Gunn
Dec 3 '18 at 21:24
By rank-nullity, $dim R + dim N = dim V$ and the question is asking about when we have a direct sum decomposition $V = R oplus N$.
– Trevor Gunn
Dec 3 '18 at 21:24
add a comment |
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You can use that the sum of the dimension of the range and of the null space is equal to the dimension of $V$. You then have two subspaces which by assumption do not intersect and whose sum of dimensions equals the dimension of the space.
– Eric
Dec 3 '18 at 21:24
By rank-nullity, $dim R + dim N = dim V$ and the question is asking about when we have a direct sum decomposition $V = R oplus N$.
– Trevor Gunn
Dec 3 '18 at 21:24