Exercises 4
[[4-1]] Β Suppose π€, π§ β π. Verify the following equalities and inequalities.
1. (a) Β π§+π§=2Reπ§
2. (b) Β π§βπ§=2(Imπ§)π
3. (c) Β π§π§ = |π§|2
4. (d) Β π€+π§=π€+π§andπ€π§=π€π§
5. (e) Β π§=π§
6. (f) Β |Reπ§| β€ |π§| and |Imπ§| β€ |π§|
7. (g) Β β£π§β£ = |π§|
8. (h) Β |π€π§| = |π€| |π§|
The results above are the parts of 4.4 that were left to the reader.
[[4-2]] Β Prove that if π€,π§ β π, then β£|π€|β|π§|β£ β€ |π€βπ§|.
The inequality above is called the reverse triangle inequality.
[[4-3]] Β Suppose π is a complex vector space and π β πβ². Define πβΆπβπ by
π(π£) = Reπ(π£) for each π£ β π.
Show that π(π£) = π(π£) β ππ(ππ£) for all π£ β π.
[[4-4]] Β Suppose π is a positive integer. Is the set
{0} βͺ {π β π«(π
) βΆ deg π = π}
a subspace of π«(π
)?
[[4-5]] Is the set
{0} βͺ {π β π«(π
) βΆ deg π is even}
a subspace of π«(π
)?
[[4-6]] Β Suppose that π and π are positive integers with π β€ π, and suppose π1, ..., ππ β π
.
Prove that there exists a polynomial π β π«(π
) with deg π = π such that 0 = π(π1) = β― = π(ππ)and such that π has no other zeros.
[[4-7]] Β Suppose that π is a nonnegative integer, π§1,...,π§π+1 are distinct elements of π
, and π€1,...,π€π+1 β π
.
Prove that there exists a unique polynomial π β π«π(π
) such that π(π§π) = π€π
This result can be proved without using linear algebra.
However, try to find for each π = 1, ..., π + 1. the clearer, shorter proof that uses some linear algebra.
[[4-8]] Β Suppose π β π«(π) has degree π. Prove that π has π distinct zeros if and only if π and its derivative πβ² have no zeros in common.
[[4-9]] Β Prove that every polynomial of odd degree with real coefficients has a real zero.
[[4-10]] Β For π β π«(π),define πβΆπβπ by
β§{π(π₯) β π(3) if π₯ =ΜΈ 3,
(ππ)(π₯) = {β¨ π₯ β 3
{β©πβ²(3) if π₯ = 3
for each π₯ β π. Show that ππ β π«(π) for every polynomial π β π«(π) and also show that πβΆ π«(π) β π«(π) is a linear map.
[[4-11]] Β Suppose π β π«(π). Define πβΆπ β π by π(π§) = π(π§) π(π§).
Prove that π is a polynomial with real coefficients.
[[4-12]] Β Suppose π is a nonnegative integer and π β π«π(π) is such that there are distinct real numbers π₯0, π₯1, ..., π₯π with π(π₯π) β π for each π = 0, 1, ..., π. Prove that all coefficients of π are real.
[[4-13]] Β Suppose π β π«(π
) with π=ΜΈ0. Let π = {ππβΆπ β π«(π
)}.
(a) Show that dimπ«(π
)/π= degπ.
(b) Find a basis of π«(π
)/π.
[[4-14]] Β Suppose π, π β π«(π) are nonconstant polynomials with no zeros in common. Let π = deg π and π = deg π.
Use linear algebra as outlined below in (a)β(c) to prove that there exist π β π«πβ1(π) and π β π«πβ1(π) such that ππ + π π = 1.
1. (a) Β Define π βΆ π«πβ1(π) Γ π«πβ1(π) β π«π+πβ1(π) by
π(π, π ) = ππ + π π.
Show that the linear map π is injective.
2. (b) Β Show that the linear map π in (a) is surjective.
3. (c) Β Use (b) to conclude that there exist π β π«πβ1(π) and π β π«πβ1(π) such that ππ + π π = 1.