Linear Algebra

Section I. Definition of Vector Space 932.19 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.⎛ ⎞ ⎛ ⎞1 0{ x ⎝0⎠ + y ⎝1⎠ }0 0⎛ ⎞1{ x ⎝0⎠ }0✄✄✘✘ ✘✘ ✘ ✘✘ ✘ ✘✘✏ ✏✏✏✏✏ ⎛ ⎞ ⎛ ⎞1 0{ x ⎝0⎠ + z ⎝0⎠ }0 1❆❍✏ ❆ ✏✏✏✏ ❍❍❍ ⎛ ⎞⎛ ⎞⎛ ⎞021{ y ⎝1⎠ } { y ⎝1⎠ } { y ⎝1⎠ } . . .001❳ ❳ ❳ ❳❳ ❍❳ ❍❍❍ ❳❳❳ ❳❳ ❍ ❅ ❅❳⎛ ⎞ ⎛ ⎞ ⎛ ⎞1 0 0{ x ⎝0⎠ + y ⎝1⎠ + z ⎝0⎠ }0 0 1⎛ ⎞ ⎛ ⎞1 0{ x ⎝1⎠ + z ⎝00 1⎠ } . . .⎛ ⎞0{ ⎝0⎠ }0We have described the subspaces as spans of sets with a minimal number ofmembers and shown them connected to their supersets. Note that the subspacesfall naturally into levels — planes on one level, lines on another, etc. — accordingto how 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, andwe saw a great variety of examples. In this subsection we saw still more spaces,ones that happen to be subspaces of others. In all of the variety we’ve seen acommonality. Example 2.19 above brings it out: vector spaces and subspacesare best understood as a span, and especially as a span of a small number ofvectors. 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 b̌ 2.21 Is this a subspace of P 2 : {a 0 + a 1 x + a 2 x 2 ∣ ∣ a0 + 2a 1 + a 2 = 4}? If it is thenparametrize its description.