Linear Algebra

86 Chapter Two. Vector Spaces1.13 Example The set F = {a cos θ+b sin θ ∣ ∣ a, b ∈ R} of real-valued functionsof the real variable θ is a vector space under the operationsand(a 1 cos θ + b 1 sin θ) + (a 2 cos θ + b 2 sin θ) = (a 1 + a 2 ) cos θ + (b 1 + b 2 ) sin θr · (a cos θ + b sin θ) = (ra) cos θ + (rb) sin θinherited from the space in the prior example. (We can think of F as “the same”as R 2 in that a cos θ + b sin θ corresponds to the vector with components a andb.)1.14 Example The set{f : R → R ∣ ∣ d2 fdx 2 + f = 0}is a vector space under the, by now natural, interpretation.(f + g) (x) = f(x) + g(x) (r · f) (x) = r f(x)In particular, notice that closure is a consequence:d 2 (f + g)dx 2+ (f + g) = ( d2 fdx 2 + f) + ( d2 gdx 2 + g)andd 2 (rf)dx 2 + (rf) = r( d2 fdx 2 + f)of basic Calculus. This turns out to equal the space from the prior example —functions satisfying this differential equation have the form a cos θ + b sin θ —but this description suggests an extension to solutions sets of other differentialequations.1.15 Example The set of solutions of a homogeneous linear system in n variablesis a vector space under the operations inherited from R n . For closureunder addition, if⎛ ⎞ ⎛ ⎞v 1w 1⎜⃗v = . ⎟ ⎜⎝ . ⎠ ⃗w = . ⎟⎝ . ⎠v n w nboth satisfy the condition that their entries add to 0 then ⃗v + ⃗w also satisfiesthat condition: c 1 (v 1 + w 1 ) + · · · + c n (v n + w n ) = (c 1 v 1 + · · · + c n v n ) + (c 1 w 1 +· · · + c n w n ) = 0. The checks of the other conditions are just as routine.As we’ve done in those equations, we often omit the multiplication symbol ‘·’.We can distinguish the multiplication in ‘c 1 v 1 ’ from that in ‘r⃗v ’ since if bothmultiplicands are real numbers then real-real multiplication must be meant,while if one is a vector then scalar-vector multiplication must be meant.The prior example has brought us full circle since it is one of our motivatingexamples.