Linear Algebra

94 Chapter Two. Vector Spaceš 2.22 Decide if the vector lies in the span of the set, inside of the space.⎛ ⎞ ⎛ ⎞ ⎛ ⎞2 1 0(a) ⎝0⎠, { ⎝0⎠ , ⎝0⎠}, in R 31 0 1(b) x − x 3 , {x 2 , 2x + x 2 , x + x 3 }, in P( ) ( ) ( ) 30 1 1 0 2 0(c) , { , }, in M 2×24 2 1 1 2 32.23 Which of these are members of the span [{cos 2 x, sin 2 x}] in the vector space ofreal-valued functions of one real variable?(a) f(x) = 1 (b) f(x) = 3 + x 2 (c) f(x) = sin x (d) f(x) = cos(2x)̌ 2.24 Which of these sets spans R 3 ? That is, which of these sets has the propertythat any three-tall vector can be expressed as a suitable linear combination of theset’s elements? ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞1 0 02 1 01 3(a) { ⎝0⎠ , ⎝2⎠ , ⎝0⎠} (b) { ⎝0⎠ , ⎝1⎠ , ⎝0⎠} (c) { ⎝1⎠ , ⎝0⎠}0 0 31 0 10 0⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞1 3 −1 22 3 5 6(d) { ⎝0⎠ , ⎝1⎠ , ⎝ 0⎠ , ⎝1⎠} (e) { ⎝1⎠ , ⎝0⎠ , ⎝1⎠ , ⎝0⎠}1 0 0 51 1 2 2̌ 2.25 Parametrize each subspace’s description. Then express each subspace as aspan.(a) The subset {(a b c) ∣ a − c = 0} of the three-wide row vectors(b) This subset of M 2×2( ) a b ∣∣{ a + d = 0}cd(c) This subset of M 2×2( ) a b ∣∣{ 2a − c − d = 0 and a + 3b = 0}c d(d) The subset {a + bx + cx 3 ∣ ∣ a − 2b + c = 0} of P3(e) The subset of P 2 of quadratic polynomials p such that p(7) = 0̌ 2.26 Find a set to span the given subspace of the given space. (Hint. Parametrizeeach.)(a) the xz-plane in R⎛ ⎞3x(b) { ⎝y⎠ ∣ 3x + 2y + z = 0} in R3z⎛ ⎞x(c) { ⎜y⎟ ∣⎝ z ⎠ 2x + y + w = 0 and y + 2z = 0} in R4w(d) {a 0 + a 1 x + a 2 x 2 + a 3 x ∣ 3 a0 + a 1 = 0 and a 2 − a 3 = 0} in P 3(e) The set P 4 in the space P 4(f) M 2×2 in M 2×22.27 Is R 2 a subspace of R 3 ?̌ 2.28 Decide if each is a subspace of the vector space of real-valued functions of onereal variable.(a) The even functions {f: R → R ∣ f(−x) = f(x) for all x}. For example, twomembers of this set are f 1 (x) = x 2 and f 2 (x) = cos(x).