## School of Mathematics

LINEAR ALGEBRA代写 Candidates must insert these four sheets into their answer booklet(s) for collection at the end of the examination.

**LINEAR ALGEBRA B **

MATH10015

(Paper code MATH–10015)

**This paper contains two sections: Section A and Section B. **

**Each section should be answered in a separate answer book. **

Section A contains FIVE questions and Section B contains TWO questions.

All SEVEN answers will be used for assessment.

Calculators of an approved type (permissible for A-Level examinations) are permitted.

Candidates may bring four hand-written sheets of A4 notes written double-sided into the examination. Candidates must insert these four sheets into their answer booklet(s) for collection at the end of the examination.

On this examination, the marking scheme is indicative and is intended only as a guide to the relative weighting of the questions.

Do not turn over until instructed.

**Section A: Short Questions LINEAR ALGEBRA代写**

A1. Let *E *= *{**e*1*, e*2*}*, *B *= *{**e*1 + *e*2*, e*1 *− **e*2*}*, and let *f *: C^{2}*→ *C^{2 }be given by the formula

* *

(a) (**2 marks**) Find *M**EE *(*f*). You may assume that *f *is linear.

(b) (**4 marks**) Find *C**EB *and *C**BE *and hence find *M**BB*(*f*). (Again, assume *f *is linear.)

(c) (**2 marks**) Compute *M**EE *(*f*) . Explain why this shows that *f *is a linear function.

### A2. Let *V *= C^{2}, a vector space over C and define **LINEAR ALGEBRA代写**

*f *: *V **× **V **→ *C*, *((*x*_{1}*, x*_{2})*,*(*y*_{1}*, y*_{2})) * **x _{1}*

*y*+

_{1}*x*

_{2}*y*

_{2}*.*

(a) (**2 marks**) Give an example of a *v*1 *∈ **V *such that *f*(*v _{1}*

*, v*)

_{1}*∈*R but

*f*(

*v*

_{1}*, v*)

_{1}*<*0.

(b) (**2 marks**) Give an example of a *v _{2}*

*∈*

*V*such that

*f*(

*v*

_{2}*, v*)

_{2}*6∈*R.

(c) (**4 marks**) Prove that for any *v, w **∈ **V *we have *f*(*v, w*) = *f*(*w, v*).

A3. Let *V *= *M _{2}*(C), the vector space of complex matrices over R, and

(a) (**4 marks**) Is the set *S *linearly independent in *V *? Justify your answer.

(b) (**2 marks**) Let *U *= span(*S*). What is dim(*U*)? What about dim(*V *)?

(c) (**2 marks**) Are *U *and *V *equal? If yes, provide some justification. If no, find an element in *V **\ **U*. **LINEAR ALGEBRA代写**

#### A4. Let F_{3}denote the field of 3 elements, let *V *= (F_{3})^{2}be a vector space over F_{3}, and define

*U *:= span*{**e*_{1}*}*.

(a) (**3 marks**) List the elements of *V *, and indicate which of these lie in *U*.

(b) (**1 mark**) State the definition of the complement of a subspace.

(c) (**4 marks**) Let *W *denote a complement of the subspace *U *in *V *. Write out each possibility for *W*. You do not have to justify that each one is a complement of *U*.

A5. Show that each of the following are not R-linear functions from *V *to *V *, where *V *= *F*(R*, *R).

(a) (**2 marks**) *ø _{1}* :

*f*(

*x*)

*f*

_{1}(

*x*), the function such that

*f*

_{1}(

*x*) = 1 for all

*x*

*∈*R.

(b) (**3 marks**) *ø _{2}* :

*f*(

*x*)

*(*

*f*(

*x*))

^{2}.

(c) (**3 marks**) *ø _{3}* :

*f*(

*x*)

*sin(*

*f*(

*x*)).

**Section B: Longer Questions LINEAR ALGEBRA代写**

**Please use a new answer book. **

B1. Let *V *= C^{3} over C, *f *: *V **→ **V *, *E *= *{**e*_{1}*, e*_{2}*, e*_{3}*} *and

(a) (**8 marks**) Find the characteristic polynomial of *A*.

(b) (**1 mark**) Find the eigenvalues of *A*. Note that these should all be integers.

(c) (**12 marks**) Find the corresponding eigenvectors for your eigenvalues in (b).

(d) (**1 mark**) Find a basis *B *such that *M**BB*(*f*) will be a diagonal matrix.

(e) (**2 marks**) Show that *A *is Hermitian.

(f) (**2 marks**) What condition could we impose on the eigenvectors in (c) to obtain a unitary matrix *U *where *U **∗**AU *is diagonal? (Note: the notation *U **∗ *denotes the adjoint matrix of *U*.)

(g) (**4 marks**) Apply your condition from (f) to your eigenvectors from (c) and check that your resulting matrix *U *is unitary, i.e., that *U **∗ *is the inverse of *U*.

### B2. Let *V *= *F*(R*, *R), a vector space over R. For each of the following questions, carefully justify your answers.

(a) (**4 marks**) For any *a **∈ *R, let *f**a*(*x*) := *a *for all *x **∈ *R. Let *S *:= *{**f**a *: *a **∈ *R*}*. LINEAR ALGEBRA代写

(i) Explain why *S *is a subset of *F*(R*, *R).

(ii) Is *S *a proper subset? If yes, find an element in *F*(R*, *R) *\ **S*.

(b) (**13 marks**) We continue with the subset *S *:= *{**f**a *: *a **∈ *R*} *of *V *= *F*(R*, *R) over R.

(i) Given *a, b **∈ *R, find *f**a *+ *f**b*. Express your answer as an element of *S*.

(ii) Does *S *contain the additive identity of *F*(R*, *R)?

(iii) Given *f **∈ **F*(R*, *R), state the key defining property for the additive inverse of *f *(which we denote by *−**f*). Is it the case, for any *a **∈ *R, that *−**f**a **∈ **S*?

(iv) Given *a, λ **∈ *R, find *λf**a*. Express your answer as an element of *S*.

(v) Is *S *a subspace of *V *?

(c) (**5 marks**) Find a non-empty subset *S*1 *⊆ **S *that is closed under addition but is not a subspace of *F*(R*, *R).

(d) (**2 marks**) Is it the case that any such *S*1 contains the zero element of *F*(R*, *R)?

(e) (**6 marks**) Does there exist a non-empty subset *S*2 *⊆ **S *that is not a subspace of *F*(R*, *R) but is closed under scalar multiplication?

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