Linear Algebra

Section I. Definition of Vector Space 972.19 Example These are the subspaces of R 3 that we now know of, the trivialsubspace, the lines through the origin, the planes through the origin, and thewhole space (of course, the picture shows only a few of the infinitely manysubspaces). In the next section we will prove that R 3 has no other type ofsubspaces, so in fact this picture shows them all.{x( 100{x( 100))+ y( 010✄✄})}{x( 100✘✘✘✘✘✘✘✘ ✘ ✘✏ ✏✏✏✏✏ ) ) ) ){x( 100+ z( 001}{x( 110❆✏ ❆ ✏✏✏✏❍ ❍❍❍ ( 0) ( 2) ( 1){y 1 } {y 1 } {y 1 } . . .001 )+ z( 001+ y( 010)+ z( 001} . . .)}❍ ❍❍❍ ❅❳❳❳❳❳❳❳❳❳❳ ❍ ❅❳ ❳ {The subsets are described as spans of sets, using a minimal number of members,and are shown connected to their supersets. Note that these subspaces fallnaturally into levels — planes on one level, lines on another, etc. — according tohow many vectors are in a minimal-sized spanning set.So far in this chapter we have seen that to study the properties of linearcombinations, the right setting is a collection that is closed under these combinations.In the first subsection we introduced such collections, vector spaces,and we saw a great variety of examples. In this subsection we saw still morespaces, ones that happen to be subspaces of others. In all of the variety we’veseen a commonality. Example 2.19 above brings it out: vector spaces and subspacesare best understood as a span, and especially as a span of a small numberof vectors. The next section studies spanning sets that are minimal.Exerciseš 2.20 Which of these subsets of the vector space of 2 × 2 matrices are subspacesunder the inherited operations? For each one that is a subspace, parametrize itsdescription. ( For ) each that is not, give a condition that fails.a 0 ∣∣(a) { a, b ∈ R}0 b( )a 0 ∣∣(b) { a + b = 0}0 b( )a 0 ∣∣(c) { a + b = 5}0 b( )a c ∣∣(d) { a + b = 0, c ∈ R}0 b( 000)}