Abstract Linear Algebra

elementary matrix

a square matrix that you get by performing a single elementary row operation on an identity matrix

3 elementary row operation

1. Add a multiple of one row to another row : R_i → Ri + cR_j (i ≠ J ; c can be any scalar)

2. Multiply a row by a nonzero scalar : R_i →cR_i (곱하기는 더하기의 연장선)

3. Switch two rows : R_i ←> R_j

matrix A is invertible (A ∈ Mₙ(𝔽))

1. L_A is bijective ⇔ L_A is isomorphism

   1. L_A is injective

      1. Ax̄ = 0̄의 해는 x̄ = 0̄뿐

   2. L_A is surjective

      1. 모든 b̄에 대해 Ax̄ = b̄가 해를 가짐

   3. 모든 b̄에 대해 Ax̄ = b̄의 해가 유일함 (해는 x̄ = A⁻¹b̄, Lec 13)

2. rank(A) = n

3. nullity(A) = 0

4. L_A sends bases to bases

5. A의 열들이 linearly independent

6. A의 열들이 𝔽ⁿ을 span, 즉 C(A) = 𝔽ⁿ (Lec 6: R(A) = column space)

7. A의 열들이 𝔽ⁿ의 basis (Lec 5: n차원 공간에서 n개 벡터는 independent ⇔ spanning ⇔ basis)

8. RREF(A) = Iₙ (Lec 13 Corollary)

9. A is a product of elementary matrices (Lec 13: RREF(A) = I이면 A = E₁⁻¹⋯Eₖ⁻¹)

[S]_{A,B}의 의미

S:V→W가 linear transformation 이고, A는 V의 bases, B는 W의 bases라고 할때, [S]_{A,B}는 입력은 A-좌표로 받고, 출력은 B-좌표로 내놓는 행렬이다.

쉽게 말해서 A의 차원의 것을 B 차원의 좌표로 나타내는 법

Dimension of a vector space V

If V is a non-zero finite dimensional vector space and A (v₁, v₂ …, v_n) is a basis of V, then we call n the dimension of V

dim({0})

0

Rank-Nullity theorem

Suppose V is a finite-dimensional vector space, and T : v→w is linear. Then

dimN(T) + dimR(T) = dimV

A linear map T : V→W is a (linear) isomorphism if ~ (definition)

There exists a inverse linear map S : W→V such that

S(T(v)) = v for all v in V and T(S(w)) = w for all w in W

A linear map T : V → W is an isomorphism <=>

T is bijective

B is a basis of vector space V <=>

every vector v in V can be written in 1 and only 1 way as a linear combination of B

two finite dimensional vector spaces are isomorphic iff ~

they have the same dimension