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# 数学考试代考 Linear algebra代写 group代写 Math 541代写

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## Mid Term Test

### Remarks.

• Do only five of the questions below.
• Definition (subsections (a)) is just a definition and there is no need to justify it. So just write it down.
• It is recommended that answers to subsections (b) and (c) are written in the following format:

0) If it is a computation start by writing the answer.

i) Main points that will appear in your explanation or proof or computation.

ii) The actual explanation or proof or computation.

The test

### 1. Linear algebra. 数学考试代考

(a) (4) Suppose V is a finite dimensional vector space over R, with an inner product 〈 ; 〉. For a subspace W  V , define its orthogonal complement W ⊂V .

(b) (10) Explain what does it mean for vectors w1 , … , wm ∊ W, to be orthogonal basis. Use such basis to show that:

1. dim(V ) = dim(W) + dim(W):

2. V = W ⊕ W, where ⊕ denotes direct sum of two spaces.

### 2. Notion of a group.

(a) (4) Define the notion of a group.

(b) (10) Let X be a set. Show that

Aut(X) = σ : X → X; σ is invertibleg.

is in a natural way a group, with the operation of composition ∘ of functions, and with identity the map id(x) = x for every x ∊ X :

### 3. Notion of subgroup. 数学考试代考

(a) (4) Suppose G is a group and H ⊂ G, a subset. Define when we say that H is a subgroup of G.

(c) (6) Consider the group (Z , + , 0) of integers. Show that if H < Z, then there is an integer d ≥ 0, such that H = dZ = {dk ; k ∊ Z }. (Show that, if H ≠ 0 ; then the smallest d > 0, that belong to H does the job. Recall also that for such d > 0, if n > 0, then there exist unique integers q ; r≥0, with 0 ≤ r < d, such that n = qd + r.).

### 4. Notions of group acting on a set.

(a) (4) Define the notion of a group G acting on a set X.

### 5. The orthogonal group O2. 数学考试代考

(a) (4) Define the orthogonal group O2 < GL2(R).

### 6. The finite subgroups of the orthogonal group O2:

(a) (4) Define when two groups (G , *G  ,1G) and (H , *H  , 1H) are isomorphic.

(b) (10) Prove the following theorem:

Theorem. Suppose G is a finite subgroup of the orthogonal group O2. We have,

### 7. Notions of homomorphism. Suppose G and H are groups. 数学考试代考

(a) (4) Define when a map φ : G →H is an homomorphism.

(b) (10) Suppose φ : G →H is an homorphism. Consider the collection

Ker(φ) = {g ∊ G ; φ(g) = 1H } ⊂ G,

called the kernel of φ.

Show that:

1. Ker(φ) a subgroup of G.

2. φis one-to-one if and only if Ker(φ) ={1G } .

(c) (6) Denote by C* the collection of all non-zero complex numbers. It is a group under usual multiplication · of complex numbers. In addition consider the collection R of real numbers. It is a group under the usual addition + of real numbers.

Consider the mapping

φ : R→C* ,

given by φ(θ) = e2πiθ = cos(2πθ) + isin(2πθ); for θ ∊ R.

1. Show that φis homorphism.

2. Show that Ker(φ) = Z and Im(φ) = S1 = {z ∊ C* ; ｜z｜= 1}the unit circle in the plane.