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Mathematics LibreTexts

20.5: Exercises

( \newcommand{\kernel}{\mathrm{null}\,}\)

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 αp(x) for αF.

2

Prove that Q(2) is a vector space.

3

Let Q(2,3) be the field generated by elements of the form a+b2+c3+d6, where a,b,c,d are in Q. Prove that Q(2,3) is a vector space of dimension 4 over Q. Find a basis for Q(2,3).

4

Prove that the complex numbers are a vector space of dimension 2 over R.

5

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

6

Let F be a field and denote the set of n-tuples of F by Fn. Given vectors u=(u1,,un) and v=(v1,,vn) in Fn and α in F, define vector addition by

u+v=(u1,,un)+(v1,,vn)=(u1+v1,,un+vn)

and scalar multiplication by

αu=α(u1,,un)=(αu1,,αun).

Prove that Fn is a vector space of dimension n under these operations.

7

Which of the following sets are subspaces of R3? If the set is indeed a subspace, find a basis for the subspace and compute its dimension.

  1. {(x1,x2,x3):3x12x2+x3=0}
  2. {(x1,x2,x3):3x1+4x3=0,2x1x2+x3=0}
  3. {(x1,x2,x3):x12x2+2x3=2}
  4. {(x1,x2,x3):3x12x22=0}

8

Show that the set of all possible solutions (x,y,z)R3 of the equations

Ax+By+Cz=0Dx+Ey+Cz=0

form a subspace of R3.

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 (αv)=(α)v=α(v) for all αF and all vV.

11

Let V be a vector space of dimension n. Prove each of the following statements.

  1. If S={v1,,vn} is a set of linearly independent vectors for V, then S is a basis for V.
  2. If S={v1,,vn} spans V, then S is a basis for V.
  3. If S={v1,,vk} is a set of linearly independent vectors for V with k<n, then there exist vectors vk+1,,vn such that

    {v1,,vk,vk+1,,vn}

    is a basis for V.

12

Prove that any set of vectors containing 0 is linearly dependent.

13

Let V be a vector space. Show that {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>n.

15. Linear Transformations

Let V and W be vector spaces over a field F, of dimensions m and n, respectively. If T:VW is a map satisfying

T(u+v)=T(u)+T(v)T(αv)=αT(v)

for all αF and all u,vV, then T is called a linear transformation from V into W.

  1. Prove that the kernel of T, ker(T)={vV:T(v)=0}, is a subspace of V. The kernel of T is sometimes called the null space of T.
  2. Prove that the range or range space of T, R(V)={wW:T(v)=w for some vV}, is a subspace of W.
  3. Show that T:VW is injective if and only if ker(T)={0}.
  4. Let {v1,,vk} be a basis for the null space of T. We can extend this basis to be a basis {v1,,vk,vk+1,,vm} of V. Why? Prove that {T(vk+1),,T(vm)} is a basis for the range of T. Conclude that the range of T has dimension mk.
  5. Let dimV=dimW. Show that a linear transformation T:VW is injective if and only if it is surjective.

16

Let V and W be finite dimensional vector spaces of dimension n over a field F. Suppose that T:VW is a vector space isomorphism. If {v1,,vn} is a basis of V, show that {T(v1),,T(vn)} is a basis of W. Conclude that any vector space over a field F of dimension n is isomorphic to Fn.

17. Direct Sums

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 uU and vV.

  1. Prove that U+V and UV are subspaces of W.
  2. If U+V=W and UV=0, then W is said to be the direct sum. In this case, we write W=UV. Show that every element wW can be written uniquely as w=u+v, where uU and vV.
  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 nk such that W=UV. Is the subspace V unique?
  4. If U and V are arbitrary subspaces of a vector space W, show that

    dim(U+V)=dimU+dimVdim(UV).

18. Dual Spaces

Let V and W be finite dimensional vector spaces over a field F.

  1. Show that the set of all linear transformations from V into W, denoted by \Hom(V,W), is a vector space over F, where we define vector addition as follows:

    (S+T)(v)=S(v)+T(v)(αS)(v)=αS(v),

    where S,T\Hom(V,W), αF, and vV.

  2. Let V be an F-vector space. Define the dual space of V to be V=\Hom(V,F). Elements in the dual space of V are called linear functionals. Let v1,,vn be an ordered basis for V. If v=α1v1++αnvn is any vector in V, define a linear functional ϕi:VF by ϕi(v)=αi. Show that the ϕi's form a basis for V. This basis is called the dual basis of v1,,vn (or simply the dual basis if the context makes the meaning clear).
  3. Consider the basis {(3,1),(2,2)} for R2. What is the dual basis for (R2)?
  4. Let V be a vector space of dimension n over a field F and let V be the dual space of V. Show that each element vV gives rise to an element λv in V and that the map vλv is an isomorphism of V with V.

This page titled 20.5: Exercises is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Thomas W. Judson (Abstract Algebra: Theory and Applications) via source content that was edited to the style and standards of the LibreTexts platform.

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