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6.E: Exercises for Chapter 6

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

Calculational Exercises

1. Define the map T:R2R2 by T(x,y)=(x+y,x).

  1. Show that T is linear.
  2. Show that T is surjective.
  3. Find dim(null(T)).
  4. Find the matrix for T with respect to the canonical basis of R2.
  5. Find the matrix for T with respect to the canonical basis for the domain R2 and the basis ((1,1),(1,1)) for the target space R2.
  6. Show that the map F:R2R2 given by F(x,y)=(x+y,x+1) is not linear.

2. Let TL(R2) be defined by

T(xy)=(yx), for all (xy)R2.

  1. Show that T is surjective.
  2. Find dim(null(T)).
  3. Find the matrix for T with respect to the canonical basis of R2.
  4. Show that the map F:R2R2 given by F(x,y)=(x+y,x+1) is not linear.

3. Consider the complex vector spaces C2 and C3 with their canonical bases, and define SL(C3,C2) be the linear map defined by S(v)=Av,vC3, where A is the matrix

A=M(S)=(i112i11).

Find a basis for null(S).

4. Give an example of a function f:R2R having

the property that

aR,vR2,f(av)=af(v)

but such that f is not a linear map.

5. Show that the linear map T:F4F2 is surjective if

null(T)={(x1,x2,x3,x4)F4 | x1=5x2,x3=7x4}.

6. Show that no linear map T:F5F2 can

have as its null space the set

{(x1,x2,x3,x4,x5)F5 | x1=3x2,x3=x4=x5}.

7. Describe the set of solutions x=(x1,x2,x3)R3 of the system of equations

x1x2+x3=0x1+2x2+x3=02x1+x2+2x3=0}.

Proof-Writing Exercises

1. Let V and W be vector spaces over F with V finite-dimensional, and let U be any
subspace of V . Given a linear map SL(U,W), prove that there exists a linear map
TL(V,W) such that, for every uU,S(u)=T(u).

2. Let V and W be vector spaces over F, and suppose that TL(V,W) is injective.
Given a linearly independent list (v1,,vn) of vectors in V, prove that the
list (T(v1),,T(vn)) is linearly independent in W.

3. Let U,V, and W be vector spaces over F, and suppose that the linear maps SL(U,V)
and TL(V,W) are both injective. Prove that the composition map TS is injective.

4. Let V and W be vector spaces over F, and suppose that TL(V,W) is surjective.
Given a spanning list (v1,,vn) for V , prove that


span(T(v1),,T(vn))=W.

5. Let V and W be vector spaces over F with V finite-dimensional. Given TL(V,W),
prove that there is a subspace U of V such that

Unull(T)={0} and range(T)={T(u)|uU}.

6. Let V be a vector space over F, and suppose that there is a linear map TL(V,V)
such that both null(T) and range(T) are finite-dimensional subspaces of V . Prove that
V must also be finite-dimensional.

7. Let U,V, and W be finite-dimensional vector spaces over F with SL(U,V) and
TL(V,W). Prove that

dim(null(TS))dim(null(T))+dim(null(S)).

8. Let V be a finite-dimensional vector space over F with S,TL(V,V). Prove that
TS is invertible if and only if both S and T are invertible.

9. Let V be a finite-dimensional vector space over F with S,TL(V,V), and denote by
I the identity map on V . Prove that TS=I if and only if ST=I.


This page titled 6.E: Exercises for Chapter 6 is shared under a not declared license and was authored, remixed, and/or curated by Isaiah Lankham, Bruno Nachtergaele, & Anne Schilling.

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