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Mathematical physics

Robert Geroch

Chapter 10

Subspaces - all with Video Answers

Educators

Chapter Questions

01:15

Problem 70

Prove that, in the category of vector spaces, a linear mapping is onto if and only if it is an epimorphism. (Hint: The proof that onto implies epimorphism is easy. For the converse, consider $V \xrightarrow{\varphi} W \xrightarrow[\beta^{\prime}]{\beta} X$, and choose $X =W$ and $\beta=i_W$. Let $P=\varphi[V]$, and let $Q$ be a complementary subspace. Let $\beta^{\prime}$ agree with $\beta$ on $P$, and let $\beta^{\prime}(q)=0$ for $q$ in $Q$.)

Rakvi .
Rakvi .
Numerade Educator

Problem 71

Let $W$ be a subspace of vector space $V$. Prove that any basis for $W$ is a subset of some basis for $V$. Obtain, from a basis for $V$ so obtained, a basis for $V / W$.

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Problem 72

Let $U$ and $U^{\prime}$ be complementary subspaces of $V$, and let $S, S^{\prime}$, and $T$ be sets the free vector spaces over which are isomorphic to $U, U^{\prime}$, and $V$, respectively. Prove that $T$ is isomorphic to $S \cup_d S^{\prime}$ (disjoint union).

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Problem 73

Let $W$ be a fixed subspace of vector space $V$. Let $U$ be a complementary subspace, and let $\psi$ be a linear mapping from $U$ to $W$. Let $U^{\prime}$ consist of all elements of $V$ which can be written in the form $u+\psi(u)$, with $u$ in $U$. Prove that $U^{\prime}$ is a subspace and is, in fact, a complementary subspace to $W$. (Thus complementary subspaces are not, except in degenerate cases, unique.) Prove, furthermore, that every complementary subspace to $W$ can be obtained, from $U$, by this construction.

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Problem 74

Let $K, L$, and $M$ be subsets of vector space $V$, and let $K+L= M$. Does therefore $K=M-L$ ?

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02:47

Problem 75

Let $V$ be a vector space, and $U, U^{\prime}, U^{\prime \prime}$ subspaces of $V$ with $U$ contained in $U^{\prime}$ and $U^{\prime}$ contained in $U^{\prime \prime}$. Consider: $\left(U^{\prime \prime} / U\right) /\left(U^{\prime} / U\right)= U^{\prime \prime} / U^{\prime}$. State the theorem and prove it.

Nick Johnson
Nick Johnson
Numerade Educator

Problem 76

Find all vector spaces whose only subspaces are the one containing only 0 and the entire vector space.

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Problem 77

Find an example of a vector space $V$ which has a subspace (other than $V$ itself) that is isomorphic to $V$.

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