Linear Algebra

Section I. Definition of Vector Space 87̌ 1.44 (a) Prove that for any four vectors ⃗v 1 , . . . ,⃗v 4 ∈ V we can associate their sumin any way without changing the result.((⃗v 1 + ⃗v 2 ) + ⃗v 3 ) + ⃗v 4 = (⃗v 1 + (⃗v 2 + ⃗v 3 )) + ⃗v 4 = (⃗v 1 + ⃗v 2 ) + (⃗v 3 + ⃗v 4 )= ⃗v 1 + ((⃗v 2 + ⃗v 3 ) + ⃗v 4 ) = ⃗v 1 + (⃗v 2 + (⃗v 3 + ⃗v 4 ))This allows us to write ‘⃗v 1 + ⃗v 2 + ⃗v 3 + ⃗v 4 ’ without ambiguity.(b) Prove that any two ways of associating a sum of any number of vectors givethe same sum. (Hint. Use induction on the number of vectors.)1.45 Example 1.5 gives a subset of R 2 that is not a vector space, under the obviousoperations, because while it is closed under addition, it is not closed under scalarmultiplication. Consider the set of vectors in the plane whose components havethe same sign or are 0. Show that this set is closed under scalar multiplication butnot addition.1.46 For any vector space, a subset that is itself a vector space under the inheritedoperations (e.g., a plane through the origin inside of R 3 ) is a subspace.(a) Show that {a 0 + a 1 x + a 2 x ∣ 2 a0 + a 1 + a 2 = 0} is a subspace of the vectorspace of degree two polynomials.(b) Show that this is a subspace of the 2×2 matrices.( ) a b ∣∣{ a + b = 0}c 0(c) Show that a nonempty subset S of a real vector space is a subspace if and onlyif it is closed under linear combinations of pairs of vectors: whenever c 1 , c 2 ∈ Rand ⃗s 1 ,⃗s 2 ∈ S then the combination c 1 ⃗v 1 + c 2 ⃗v 2 is in S.I.2 Subspaces and Spanning SetsOne of the examples that led us to introduce the idea of a vector space was thesolution set of a homogeneous system. For instance, we’ve seen in Example 1.4such a space that is a planar subset of R 3 . There, the vector space R 3 containsinside it another vector space, the plane.2.1 Definition For any vector space, a subspace is a subset that is itself a vectorspace, under the inherited operations.2.2 Example The plane from the prior subsection,⎛ ⎞x⎜ ⎟P = { ⎝y⎠ ∣ x + y + z = 0}zis a subspace of R 3 . As specified in the definition, the operations are the onesthat are inherited from the larger space, that is, vectors add in P as they add inR 3⎛ ⎞ ⎛ ⎞ ⎛ ⎞x 1 x 2 x 1 + x 2⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝y 1 ⎠ + ⎝y 2 ⎠ = ⎝y 1 + y 2 ⎠z 1 z 2 z 1 + z 2