Chapter 1 theorems Advanced Linear algebra

If x,y, and z are vectors in a vector space V such that x+z=y+z, then x=y.
Corollary 1. The vector 0 is unique.
Corollary 2. The vector y is unique.
Theorem 1.1 (Cancellation Law for Vector Addition)
In any vector space V, the following statements are true:
a) 0x=0 for each x∈V
b)(-a)x = -(ax)= a(-x) for each a∈F and each x∈V
c) a0=0 for each a∈F
Theorem 1.2
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Let V be a vector space and W a subset of V. Then W is a subspace of V if and only if the following three conditions hold for operations defined on V:
a) 0∈W
b) x+y ∈W whenever x∈W and y∈W
c) cx∈W whenever c∈F and x∈W
Theorem 1.3
Any intersection of subspaces of a vector space V is a subspace of V
Theorem 1.4
The span of any subset S of a vector space V is a subspace of V. Moreover, any subspace of V that contains S must also contain the span of S
Theorem 1.5
Let V be a vector space, and let S_1⊆S_2⊆V. If S_2 is linearly independent, then S_1 is linearly independent.
Corollary: Let V be a vector space, and let S_1⊆S_2⊆V. If S_2 is linearly dependent, then S_1 is linearly dependent.
Theorem 1.6
Let S be a linearly independent subset of a vector space V, and let v be a vector in V that is not in S. Then S∪{v} is linearly independent if and only if v∈span(S).
Theorem 1.7
If a vector space V is generated by a finite set S, then some subset of S is a basis for V. Hence V has a finite basis.
Theorem 1.9
Let V be a vector space that is generated by a set G containing exactly m vectors. Then m≤n and there exists a subset H of G containing exactly n-m vectors such that L∪H generates V.
Theorem 1.10 (Replacement Theorem)
Let V be a vector space having a finite basis. Then every basis for V contains the same number of vectors.
Corollary 1 Replacement Theorem
Let V be a vector space with dimension n.
a) Any finite generating set for V contains at least n vectors, and a generating set for V that contains exactly n vectors is a basis for V.
b) Any linearly independent subset of V that contains exactly n vectors is a basis for V
c)Every linearly independent subset of V can be extended to a basis for V.
Corollary 2 Replacement Theorem
Let W be a subspace of a finite-dimensional vector space V. Then W is finite-dimensional dim(W)≤dim(V). Moreover, if dim(W)=dim(V), then V=W
Corollary: If W is a subspace of a finite-dimensional vector space V, then any basis for W can be extended to a basis for V.
Theorem 1.11
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