Linear Algebra

Section I. Definition of Vector Space 95(b) The odd functions {f: R → R ∣ f(−x) = −f(x) for all x}.f 3 (x) = x 3 and f 4 (x) = sin(x).Two members are2.29 Example 2.16 says that for any vector ⃗v that is an element of a vector spaceV, the set {r · ⃗v ∣ ∣ r ∈ R} is a subspace of V. (This is of course, simply the span ofthe singleton set {⃗v}.) Must any such subspace be a proper subspace, or can it beimproper?2.30 An example following the definition of a vector space shows that the solutionset of a homogeneous linear system is a vector space. In the terminology of thissubsection, it is a subspace of R n where the system has n variables. What abouta non-homogeneous linear system; do its solutions form a subspace (under theinherited operations)?2.31 [Cleary] Give an example of each or explain why it would be impossible to doso.(a) A nonempty subset of M 2×2 that is not a subspace.(b) A set of two vectors in R 2 that does not span the space.2.32 Example 2.19 shows that R 3 has infinitely many subspaces. Does every nontrivialspace have infinitely many subspaces?2.33 Finish the proof of Lemma 2.9.2.34 Show that each vector space has only one trivial subspace.̌ 2.35 Show that for any subset S of a vector space, the span of the span equals thespan [[S]] = [S]. (Hint. Members of [S] are linear combinations of members of S.Members of [[S]] are linear combinations of linear combinations of members of S.)2.36 All of the subspaces that we’ve seen use zero in their description in some way.For example, the subspace in Example 2.3 consists of all the vectors from R 2 witha second component of zero. In contrast, the collection of vectors from R 2 with asecond component of one does not form a subspace (it is not closed under scalarmultiplication). Another example is Example 2.2, where the condition on thevectors is that the three components add to zero. If the condition were that thethree components add to one then it would not be a subspace (again, it would failto be closed). This exercise shows that a reliance on zero is not strictly necessary.Consider the set⎛ ⎞x{ ⎝y⎠ ∣ x + y + z = 1}zunder these operations.⎛ ⎛ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞x 1 + x 2 − 1 x rx − r + 1⎞ ⎞x 1 x 2⎝y 1⎠ + ⎝y 2⎠ = ⎝z 1 z 2y 1 + y 2z 1 + z 2⎠r ⎝ ⎠ = ⎝(a) Show that it is not a subspace of R 3 . (Hint. See Example 2.5).(b) Show that it is a vector space. Note that by the prior item, Lemma 2.9 cannot apply.(c) Show that any subspace of R 3 must pass through the origin, and so anysubspace of R 3 must involve zero in its description. Does the converse hold?Does any subset of R 3 that contains the origin become a subspace when giventhe inherited operations?2.37 We can give a justification for the convention that the sum of zero-many vectorsequals the zero vector. Consider this sum of three vectors ⃗v 1 +⃗v 2 +⃗v 3 .(a) What is the difference between this sum of three vectors and the sum of thefirst two of these three?yzryrz⎠