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Math 541

HW1 – Linear Algebra Refresher

数学线性代数代写 Remarks: A) Definition is just a definition, there is no need to justify or explain it. B) Answers to questions with proofs should be written

Remarks:

A) Definition is just a definition, there is no need to justify or explain it.

B) Answers to questions with proofs should be written in the following format:

i) Statement and/or Result.

ii) Main points that will appear in your proof.

iii) The actual proof.

C) Answers to questions with computations should be written in the following format:

i) Statement and/or Result.

ii) Main points that will appear in your computation.

iii) The actual computation.

1. The field C of complex numbers. 数学线性代数代写

(a) Define the field C, namely write how its elements look like, and how to multiply any two complex numbers.

(b) Fix an integer n ≥ 1. Write down the standard formula for all complex numbers z ∊ C, that satisfy

zⁿ = 1. (1)

(c) If you think on C as the plane R² , and draw there the points z that solve the equation (1), an connect any two adjacent points by a line, you obtain a configuration called n-gon, denoted Gn. Draw the n-gon G_n; for n = 3，4，5.

2. Notions of vector space, basis, and dimension. 数学线性代数代写

(a) Do the following:

1. Write down the definition of a vector space V over a field F.

2. Suppose V is a vector space over a field F. Define when a subset B ⊂ V is called basis of V .

3. Define when a vector space V over a field F is finite-dimensional, and, in this case, define its dimension.

(c) Recall what is the finite field F₂ with two elements (i.e., write down its elements and the addition and multiplication table). Now, suppose n is a non-negative integer, and V is a vector space of dimension n over the field F₂. How many elements V has? Explain your answer.

3. Linear transformations.

(a) Suppose V and W are vector spaces over a field F. Define when a map

T : V → W,

is called linear transformation.

(b) Suppose V is an n-dimensional vector space over a field F, and B = {v₁， …， v_n} is a basis of V . In addition, suppose T : V → V， is a linear transformation. Denote by M_n = M_n(F) the set of n × n matrices with entries from F.

4. Inner product spaces. In this section, F denotes the fields C or R, and V is a vector space over F. 数学线性代数代写

(a) Define what is an inner product〈，〉 on V . In this case the pair (V，〈，〉) is called inner product space.

(b) Suppose that V is finite dimensional, with inner product 〈，〉. Suppose W ⊂ V is a subspace. The subspace of V given by,

W^⊥ = {v ∊ V ; 〈v ， w〉 = 0, for every w ∊ W}，

is called the orthogonal complement of W. Do the following:

1. dim(W) + dim(W^⊥) = dim(V ):

2. For two subspaces V₁; V₂ ⊂ V , define when V is a direct sum of V₁ and V₂, denoted V = V₁ ⊕ V₂. Show that in our case,

V = W ⊕ W^⊥. 

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