Linear Algebra

114 Chapter Two. Vector Spaces1.18 Find a basis for P 2 , the space of all quadratic polynomials. Must any suchbasis contain a polynomial of each degree: degree zero, degree one, and degree two?1.19 Find a basis for the solution set of this system.x 1 − 4x 2 + 3x 3 − x 4 = 02x 1 − 8x 2 + 6x 3 − 2x 4 = 0̌ 1.20 Find a basis for M 2×2 , the space of 2×2 matrices.̌ 1.21 Find a basis for each.∣(a) The subspace {a 2 x 2 + a 1 x + a 0 a2 − 2a 1 = a 0 } of P 2(b) The space of three-wide row vectors whose first and second components addto zero(c) This subspace of the 2×2 matrices( ) a b ∣∣{ c − 2b = 0}0 c1.22 Check Example 1.6.̌ 1.23 Find the span of each set and then find a basis for that span.(a) {1 + x, 1 + 2x} in P 2 (b) {2 − 2x, 3 + 4x 2 } in P 2̌ 1.24 Find a basis for each of these subspaces of the space P 3 of cubic polynomials.(a) The subspace of cubic polynomials p(x) such that p(7) = 0(b) The subspace of polynomials p(x) such that p(7) = 0 and p(5) = 0(c) The subspace of polynomials p(x) such that p(7) = 0, p(5) = 0, and p(3) = 0(d) The space of polynomials p(x) such that p(7) = 0, p(5) = 0, p(3) = 0,and p(1) = 01.25 We’ve seen that the result of reordering a basis can be another basis. Must itbe?1.26 Can a basis contain a zero vector?̌ 1.27 Let 〈⃗β 1 , ⃗β 2 , ⃗β 3 〉 be a basis for a vector space.(a) Show that 〈c 1⃗β 1 , c 2⃗β 2 , c 3⃗β 3 〉 is a basis when c 1 , c 2 , c 3 ≠ 0. What happenswhen at least one c i is 0?(b) Prove that 〈⃗α 1 , ⃗α 2 , ⃗α 3 〉 is a basis where ⃗α i = ⃗β 1 + ⃗β i .1.28 Find one vector ⃗v that will make each into a basis for the space.⎛ ⎞ ⎛ ⎞( 1 01(a) 〈 ,⃗v〉 in R1)2 (b) 〈 ⎝1⎠ , ⎝1⎠ ,⃗v〉 in R 3 (c) 〈x, 1 + x 2 ,⃗v〉 in P 20 0̌ 1.29 Where 〈⃗β 1 , . . . , ⃗β n 〉 is a basis, show that in this equationeach of the c i ’s is zero. Generalize.c 1⃗β 1 + · · · + c k⃗β k = c k+1⃗β k+1 + · · · + c n⃗β n1.30 A basis contains some of the vectors from a vector space; can it contain themall?1.31 Theorem 1.12 shows that, with respect to a basis, every linear combination isunique. If a subset is not a basis, can linear combinations be not unique? If so,must they be?̌ 1.32 A square matrix is symmetric if for all indices i and j, entry i, j equals entryj, i.(a) Find a basis for the vector space of symmetric 2×2 matrices.(b) Find a basis for the space of symmetric 3×3 matrices.(c) Find a basis for the space of symmetric n×n matrices.