Linear Algebra

84 Chapter Two. Vector SpacesWarning! The examples so far involve sets of column vectors with the usualoperations. But vector spaces need not be collections of column vectors, or evenof row vectors. Below are some other types of vector spaces. The term ‘vectorspace’ does not mean ‘collection of columns of reals’. It means something morelike ‘collection in which any linear combination is sensible’.1.8 Example Consider P 3 = {a 0 + a 1 x + a 2 x 2 + a 3 x 3 ∣ ∣ a 0 , . . . , a 3 ∈ R}, theset of polynomials of degree three or less (in this book, we’ll take constantpolynomials, including the zero polynomial, to be of degree zero). It is a vectorspace under the operationsand(a 0 + a 1 x + a 2 x 2 + a 3 x 3 ) + (b 0 + b 1 x + b 2 x 2 + b 3 x 3 )= (a 0 + b 0 ) + (a 1 + b 1 )x + (a 2 + b 2 )x 2 + (a 3 + b 3 )x 3r · (a 0 + a 1 x + a 2 x 2 + a 3 x 3 ) = (ra 0 ) + (ra 1 )x + (ra 2 )x 2 + (ra 3 )x 3(the verification is easy). This vector space is worthy of attention because theseare the polynomial operations familiar from high school algebra. For instance,3 · (1 − 2x + 3x 2 − 4x 3 ) − 2 · (2 − 3x + x 2 − (1/2)x 3 ) = −1 + 7x 2 − 11x 3 .Although this space is not a subset of any R n , there is a sense in which wecan think of P 3 as “the same” as R 4 . If we identify these two spaces’s elementsin this way⎛ ⎞a 0a 0 + a 1 x + a 2 x 2 + a 3 x 3 corresponds to ⎜a 1⎟⎝a 2⎠a 3then the operations also correspond. Here is an example of corresponding additions.⎛ ⎞ ⎛ ⎞ ⎛ ⎞1 − 2x + 0x 2 + 1x 31+ 2 + 3x + 7x 2 − 4x 3 corresponds to ⎜−2⎟⎝ ⎠ + ⎜ ⎟⎝ ⎠ = ⎜ ⎟⎝ ⎠3 + 1x + 7x 2 − 3x 3Things we are thinking of as “the same” add to “the same” sum. Chapter Threemakes precise this idea of vector space correspondence. For now we shall justleave it as an intuition.1.9 Example The set M 2×2 of 2×2 matrices with real number entries is avector space under the natural entry-by-entry operations.( ) ( ) ( ) ( ) ( )a b w x a + w b + x a b ra rb+ =r · =c d y z c + y d + z c d rc rd01237−4As in the prior example, we can think of this space as “the same” as R 4 .317−3