Vector Spaces: exercises

Practice (16)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

1. If \(F\) is a field, show that \(F[x]\) is a vector space over \(F\), where the vectors in \(F[x]\) are polynomials. Vector addition is polynomial addition, and scalar multiplication is defined by \(\alpha p(x)\) for \(\alpha \in F\).

2. Prove that \({\mathbb Q }( \sqrt{2}\, )\) is a vector space.

3. Let \({\mathbb Q }( \sqrt{2}, \sqrt{3}\, )\) be the field generated by elements of the form \(a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6}\), where \(a, b, c, d\) are in \({\mathbb Q}\). Prove that \({\mathbb Q }( \sqrt{2}, \sqrt{3}\, )\) is a vector space of dimension \(4\) over \({\mathbb Q}\). Find a basis for \({\mathbb Q }( \sqrt{2}, \sqrt{3}\, )\).

   答えを明らかにしろ

   Hint:

   \({\mathbb Q}(\sqrt{2}, \sqrt{3}\, )\) has basis \(\{ 1, \sqrt{2}, \sqrt{3}, \sqrt{6}\, \}\) over \({\mathbb Q}\).

4. Prove that the complex numbers are a vector space of dimension \(2\) over \({\mathbb R}\).

5. Prove that the set \(P_n\) of all polynomials of degree less than \(n\) form a subspace of the vector space \(F[x]\). Find a basis for \(P_n\) and compute the dimension of \(P_n\).

   答えを明らかにしろ

   Hint:

   The set \(\{ 1, x, x^2, \ldots, x^{n-1} \}\) is a basis for \(P_n\).

6. Let \(F\) be a field and denote the set of \(n\)-tuples of \(F\) by \(F^n\). Given vectors \(u = (u_1, \ldots, u_n)\) and \(v = (v_1, \ldots, v_n)\) in \(F^n\) and \(\alpha\) in \(F\), define vector addition by \[\begin{aligned}\end{aligned}\] and scalar multiplication by \[\begin{aligned}\end{aligned}\]. Prove that \(F^n\) is a vector space of dimension \(n\) under these operations.

7. Which of the following sets are subspaces of \({\mathbb R}^3\)? If the set is indeed a subspace, find a basis for the subspace and compute its dimension.

   1. \(\{ (x_1, x_2, x_3) : 3 x_1 - 2 x_2 + x_3 = 0 \}\)

   2. \(\{ (x_1, x_2, x_3) : 3 x_1 + 4 x_3 = 0, 2 x_1 - x_2 + x_3 = 0 \}\)

   3. \(\{ (x_1, x_2, x_3) : x_1 - 2 x_2 + 2 x_3 = 2 \}\)

   4. \(\{ (x_1, x_2, x_3) : 3 x_1 - 2 x_2^2 = 0 \}\)

   答えを明らかにしろ

   Hint:

   (a) Subspace of dimension \(2\) with basis \(\{(1, 0, -3), (0, 1, 2) \}\); (d) not a subspace

8. Show that the set of all possible solutions \((x, y, z) \in {\mathbb R}^3\) of the equations \[\begin{aligned}Ax + B y + C z & = 0 \\ D x + E y + C z & = 0\end{aligned}\] form a subspace of \({\mathbb R}^3\).

9. Let \(W\) be the subset of continuous functions on \([0, 1]\) such that \(f(0) = 0\). Prove that \(W\) is a subspace of \(C[0, 1]\).

10. Let \(V\) be a vector space over \(F\). Prove that \(-(\alpha v) = (-\alpha)v = \alpha(-v)\) for all \(\alpha \in F\) and all \(v \in V\).

    答えを明らかにしろ

    Hint:

    Since \(0 = \alpha 0 = \alpha(-v + v) = \alpha(-v) + \alpha v\), it follows that \(- \alpha v = \alpha(-v)\).

11. Let \(V\) be a vector space of dimension \(n\). Prove each of the following statements.

    1. If \(S = \{v_1, \ldots, v_n \}\) is a set of linearly independent vectors for \(V\), then \(S\) is a basis for \(V\).

    2. If \(S = \{v_1, \ldots, v_n \}\) spans \(V\), then \(S\) is a basis for \(V\).

    3. If \(S = \{v_1, \ldots, v_k \}\) is a set of linearly independent vectors for \(V\) with \(k \lt n\), then there exist vectors \(v_{k + 1}, \ldots, v_n\) such that \[\begin{aligned}\end{aligned}\] is a basis for \(V\).

12. Prove that any set of vectors containing \({\mathbf 0}\) is linearly dependent.

    答えを明らかにしろ

    Hint:

    Let \(v_0 = 0, v_1, \ldots, v_n \in V\) and \(\alpha_0 \neq 0, \alpha_1, \ldots, \alpha_n \in F\). Then \(\alpha_0 v_0 + \cdots + \alpha_n v_n = 0\).

13. Let \(V\) be a vector space. Show that \(\{ {\mathbf 0} \}\) is a subspace of \(V\) of dimension zero.

14. If a vector space \(V\) is spanned by \(n\) vectors, show that any set of \(m\) vectors in \(V\) must be linearly dependent for \(m \gt n\).

15. Let \(V\) and \(W\) be finite dimensional vector spaces of dimension \(n\) over a field \(F\). Suppose that \(T: V \rightarrow W\) is a vector space isomorphism. If \(\{ v_1, \ldots, v_n \}\) is a basis of \(V\), show that \(\{ T(v_1), \ldots, T(v_n) \}\) is a basis of \(W\). Conclude that any vector space over a field \(F\) of dimension \(n\) is isomorphic to \(F^n\).

16. Let \(U\) and \(V\) be subspaces of a vector space \(W\). The sum of \(U\) and \(V\), denoted \(U + V\), is defined to be the set of all vectors of the form \(u + v\), where \(u \in U\) and \(v \in V\).

    1. Prove that \(U + V\) and \(U \cap V\) are subspaces of \(W\).

    2. If \(U + V = W\) and \(U \cap V = {\mathbf 0}\), then \(W\) is said to be the direct sum. In this case, we write \(W = U \oplus V\). \(U \oplus V\) direct sum of vector spaces \(U\) and \(V\) Show that every element \(w \in W\) can be written uniquely as \(w = u + v\), where \(u \in U\) and \(v \in V\).

    3. Let \(U\) be a subspace of dimension \(k\) of a vector space \(W\) of dimension \(n\). Prove that there exists a subspace \(V\) of dimension \(n-k\) such that \(W = U \oplus V\). Is the subspace \(V\) unique?

    4. If \(U\) and \(V\) are arbitrary subspaces of a vector space \(W\), show that \[\begin{aligned}\end{aligned}\].

    答えを明らかにしろ

    Hint:

    (a) Let \(u, u' \in U\) and \(v, v' \in V\). Then \[\begin{aligned}(u + v) + (u' + v') & = (u + u') + (v + v') \in U + V \\ \alpha(u + v) & = \alpha u + \alpha v \in U + V\end{aligned}\].

Symbols used here

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

x \in A,\ A \subseteq B

element of, subset
x belongs to A; every element of A is in B.

A \cup B,\ A \cap B,\ A \setminus B

union, intersection, difference
In either; in both; in A but not B.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

\blacksquare\ \text{or}\ \square

end of proof (halmos)
Marks the point where the statement has been established.

a \equiv b \pmod n

congruent modulo n
n divides a − b; a and b have the same remainder.

a \mid b,\ \gcd(a,b)

divides, greatest common divisor
b is a multiple of a; the largest number dividing both.

(G, \cdot),\ e,\ g^{-1}

group, identity, inverse
A set with an operation; the do-nothing element; the element that undoes g.

G \cong H,\ G / N

isomorphic, quotient group
Same structure; the group of cosets of a normal subgroup N.

\mathbb{Z}/n\mathbb{Z},\ \mathbb{Z}_n

integers modulo n
The remainders 0…n−1 with clock arithmetic.

\operatorname{Hom}(A, B),\ f \circ g

arrows from A to B, composition
The set of morphisms; do g then f.

Questions people ask

What is a group, in plain words?

A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group — that is where the idea came from.

What is the difference between a ring and a field?

A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.

あなた自身を試してみてください

⌨ Type ✎ Draw

数学の問題

解決

x²√÷×π( )=∫d/dxlim[ ]sinlogeˣ

Parts of this page are adapted from Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Condensed and re-explained here; errors are ours.

ここに Abstract Algebra

GroupsSubgroups, cosets and Lagrange's theoremCyclic groups and permutation groupsHomomorphisms, normal subgroups and quotient groupsRings and fieldsGalois theory: why the quintic has no formula