Linear Algebra

98 Chapter Two. Vector Spaceš 2.21 Is this a subspace of P 2: {a 0 + a 1x + a 2x 2 ∣ ∣ a0 + 2a 1 + a 2 = 4}? If it is thenparametrize its description.̌ 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 M4 2 1 1 2 32×22.23 Which of these are members of the span [{cos 2 x, sin 2 x}] in the vector spaceof real-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 0(a) {(0 , 2 , 00 0 3) )(d) {( 101,( 310,( −1002} (b) {(01) ),( 215,( 110} (e) {( 211( 0, 01),( 3011} (c) {(10) ) ),( 512,( 602},( 300̌ 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⎜y⎟(c) { ⎝z⎠ ∣ 2x + y + w = 0 and y + 2z = 0} in R4w∣(d) {a 0 + a 1x + a 2x 2 + a 3x 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 ?