20.5: Exercises
( \newcommand{\kernel}{\mathrm{null}\,}\)
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 α∈F.
Prove that Q(√2) is a vector space.
Let Q(√2,√3) be the field generated by elements of the form a+b√2+c√3+d√6, 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).
Prove that the complex numbers are a vector space of dimension 2 over R.
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.
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.
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.
- {(x1,x2,x3):3x1−2x2+x3=0}
- {(x1,x2,x3):3x1+4x3=0,2x1−x2+x3=0}
- {(x1,x2,x3):x1−2x2+2x3=2}
- {(x1,x2,x3):3x1−2x22=0}
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.
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].
Let V be a vector space over F. Prove that −(αv)=(−α)v=α(−v) for all α∈F and all v∈V.
Let V be a vector space of dimension n. Prove each of the following statements.
- If S={v1,…,vn} is a set of linearly independent vectors for V, then S is a basis for V.
- If S={v1,…,vn} spans V, then S is a basis for V.
- 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.
Prove that any set of vectors containing 0 is linearly dependent.
Let V be a vector space. Show that {0} is a subspace of V of dimension zero.
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.
Linear Transformations
Let V and W be vector spaces over a field F, of dimensions m and n, respectively. If T:V→W is a map satisfying
T(u+v)=T(u)+T(v)T(αv)=αT(v)
for all α∈F and all u,v∈V, then T is called a linear transformation from V into W.
- Prove that the kernel of T, ker(T)={v∈V:T(v)=0}, is a subspace of V. The kernel of T is sometimes called the null space of T.
- Prove that the range or range space of T, R(V)={w∈W:T(v)=w for some v∈V}, is a subspace of W.
- Show that T:V→W is injective if and only if ker(T)={0}.
- 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 m−k.
- Let dimV=dimW. Show that a linear transformation T:V→W is injective if and only if it is surjective.
Let V and W be finite dimensional vector spaces of dimension n over a field F. Suppose that T:V→W 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.
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 u∈U and v∈V.
- Prove that U+V and U∩V are subspaces of W.
- If U+V=W and U∩V=0, then W is said to be the direct sum. In this case, we write W=U⊕V. Show that every element w∈W can be written uniquely as w=u+v, where u∈U and v∈V.
- 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⊕V. Is the subspace V unique?
- If U and V are arbitrary subspaces of a vector space W, show that
dim(U+V)=dimU+dimV−dim(U∩V).
Dual Spaces
Let V and W be finite dimensional vector spaces over a field F.
- 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), α∈F, and v∈V.
- 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:V→F 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).
- Consider the basis {(3,1),(2,−2)} for R2. What is the dual basis for (R2)∗?
- 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 v∈V gives rise to an element λv in V∗∗ and that the map v↦λv is an isomorphism of V with V∗∗.