Read the summary and the most important questions on Elementary Linear Algebra (Classic Version) | 9780134689470 | Lawrence E. Spence; Arnold J. Insel; Stephen H. Friedberg

• 1.3 Subspaces

• What is a subspace?

  - A subspace is a subset (W) in V over a field F in which the closure properties must hold and the Ov from V must be in W as well.
  - for any x, y in W, x+y is also inW
  - for any c in F and x in W, cx is also in W
  -0v is in W

• Is the intersection of subspaces from V a subspace of V?

  Yes, 
  If, C is a number of subspaces in V 
  and W stands for the intersection of subspaces in C   
  -Then we know that Ov is in W as it is contained in every subspace in C given that they are subspaces.
  -Now, let c be a scalar from F,
  and x, y vectors from W
  then x+y are contained in each subspace in C
  and cx is also contained in each subspace in C
  Hence, x+y and cx are in W 
  -So by definition W is a subspace of V because all 3 conditions hold

• 1.4 Linear Combinations & System of Linear Equations

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• What is a Span?

  - The set of all possible linear combinations in V 
  - If S is a nonempty subset of V, V being a vector space.
  then span(S) is the set made of all linear combinations of vectors v in subset S 
  -span(empty set)= 0
  - Span of any subset (S) of a vector space (V), is a subspace of V that contains S, in other words any V that contains subset S must also contain span(S) which is a subspace of V

• What is a generating set?

  - a generating set is a subset (S) in the vector space V, whose vectors generate/spans V
  - S is a generating set/ S spans V if span(S)=V
  - in other words every vector in V can be written as a linear combination of vectors v in S, so the span(S) "produces" V

• 1.5 Linear Dependence & Independence

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• What is a Linearly Dependent Subset?

  -If S is a subset in vector space V and there exists a finite number of distinct vectors u1,u2,..,un in S
  and a finite number of distinct scalars a1,a2,...,an (NOT ALL ZERO) in V
  such that  a1u1+a2u2+...+anun=0
  then S is called linearly dependent.
  -Any subset S of vector space V that contains the zero vector is a linearly dependent subset because 0v->(1)(0) = 0 is a nontrivial representation of 0 as a linear combination of vectors in S

• What is a Linearly Independent subset?

  -A subset S of a vector space V is linearly independent iff the only representation of 0 as linear combination of vecotrs u in V are trivial representations.
  - a1u1+a2u2+...+anun= 0 -> a1=a2=...=an=0
  -the empty set is linearly independent because linearly dependent sets must be nonempty
  -a set containing only one nonzero vector is linearly independent because the only solution for a1u1=0, given that u1 is not 0 is a1=0

• What are some immediate consequences of the definitions of linear dependence & independence?

  1. If we have V, a vector space that contains S2 and S2 contains S1, if S1 is linearly dependent, then S2 is linearly dependent.
  2.If  V is a vector space that contains S2 and S2 contains S1, if S2 is linearly independent, then S1 is linearly independent.

• 1.6 Base & Dimension

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• What is a Basis?

  -it is a linearly independent subset of vector space V, that generates V.
  - "A generating set"
  -Span(B)=V -> any vector v in V can be written as a linear combination of vectors in B
  - Knowing that Span(empty set) is linearly independent (Span(emptyset)=0) so (empty set) is a basis for zero vector space
  -in Fn -> e1,e2,e3,...are standard basis for Fn e1=(1,0,...) e2=(0,1,0..)...
  -in Pn(R) -> {1,x,x^2,...} is standard basis for polynomials

• What is the Replacement Theorem?

  Repalement theorem defines a generating set LuH for vector space V. Where V is a vector space that is generated by a set G which contains n vectors. And where L is a linearly independent subset of V which contains m vectors.
  Then m</=n and there exists a subset H of G containing n-m vectors such than LuH generates V
  
  -If V is a vector space with finite basis, then all bases for V are finite and contain the exact same number of vectors
  -> if B,Y,Q are bases for V then they all contain n vectors.

• What is a Finite Dimensional Vector Space?

  -It is a vector space which has a finite number of n vectors in its basis.
   - The number n of elements which is the same for every basis of V is called the dimension of V, dim(V)
  -A vector space with an infinite number of vectors in its basis is called an infinite dimensional vector space
  -{0} has dimension 0
  - F^n has dimension n
  -Mmxn has dimension mn
  -Pn(F) has dimension n+1
  -P(F) is infinite dimensional because it's basis is {1,x,x^2,......}