Summary linear algebra (notes, exercises, correction)

CHAPTER 4 VECTOR SPACES

4.1 Vector Spaces and Subspaces


• Vector Space

A vector space is a nonempty set V of objects, called vectors, on which are
defined two operations, called addition and multiplication by scalars, subject
to the following axioms.
1. The sum of u and v, denoted by u + v is in V .
2. u + v = v + u.
3. (u + v) + w = u + (v + w).
4. There is a zero vector 0 in V such that u + 0 = u.
5. For each u in V , there is a vector −u in V such that u + (−u) = 0.
6. For any c ∈ R and u ∈ V , cu is in V .
7. c (u + v) = cu + cv.
8. (c + d)u = cu + du.
9. c(du) = (cd)u.
10. 1u = u.


• Example Rn is a vector space.




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• Example The set of all infinite sequences (c0 , c1 , c2 , . . . , ).




• Example Pn – the set of all polynomials of degree at most n.




• Example The set of all real-valued functions defined on a set D.

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• Theorem

• The zero vector is unique.
• The negative of a vector is unique.
• 0u = 0
• c0 = 0
• −u = (−1)u

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• Subspaces

A subspace of a vector space V is a subset H of V that is itself a vector
space. It should have the following three properties.
• The zero vector is in H.
• If u and v are in H, then u + v is in H.
• For any c ∈ R and u ∈ H, cu is in H.



• Example {0} is a subspace of any vector space, called the zero subspace.


• Example If m < n, Pm is a subspace of Pn .

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• Example H =  t  : s, t ∈ R is a subspace of R3 .
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• Example H =  t  : s, t ∈ R is not a subspace of R3 .
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