Linear Algebra

Section I. Definition of Vector Space 85(d) Under the usual polynomial operations,{a 0 + a 1 x + a 2 x 2 ∣ ∣ a0 , a 1 , a 2 ∈ R + }where R + is the set of reals greater than zero(e) Under the inherited operations,( x{ ∈ Ry)∣ 2 x + 3y = 4 and 2x − y = 3 and 6x + 4y = 10}1.23 Define addition and scalar multiplication operations to make the complexnumbers a vector space over R.̌ 1.24 Is the set of rational numbers a vector space over R under the usual additionand scalar multiplication operations?1.25 Show that the set of linear combinations of the variables x, y, z is a vector spaceunder the natural addition and scalar multiplication operations.1.26 Prove that this is not a vector space: the set of two-tall column vectors withreal entries subject to these operations.( ) ( ) ( ) ( ( )x1 x2 x1 − x 2x rx+ =r · =y 1 y 2 y 1 − y 2 y)ry1.27 Prove or disprove that R 3 is a vector space under these operations.⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞x 1 x 2 0x rx(a) ⎝y 1⎠ + ⎝y 2⎠ = ⎝0⎠ and r ⎝y⎠ = ⎝ry⎠z 1 z 2 0z rz⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞x 1 x 2 0x 0(b) ⎝y 1⎠ + ⎝y 2⎠ = ⎝0⎠ and r ⎝y⎠ = ⎝0⎠z 1 z 2 0z 0̌ 1.28 For each, decide if it is a vector space; the intended operations are the naturalones.(a) The diagonal 2×2 matrices( ) a 0 ∣∣{ a, b ∈ R}0 b(b) This set of 2×2 matrices( ) x x + y ∣∣{x, y ∈ R}x + y y(c) This set⎛ ⎞x{ ⎜y⎟⎝ z ⎠ ∈ ∣ R4 x + y + w = 1}w(d) The set of functions {f: R → R ∣ ∣ df/dx + 2f = 0}(e) The set of functions {f: R → R ∣ ∣ df/dx + 2f = 1}̌ 1.29 Prove or disprove that this is a vector space: the real-valued functions f of onereal variable such that f(7) = 0.̌ 1.30 Show that the set R + of positive reals is a vector space when we interpret ‘x + y’to mean the product of x and y (so that 2 + 3 is 6), and we interpret ‘r · x’ as ther-th power of x.1.31 Is {(x, y) ∣ ∣ x, y ∈ R} a vector space under these operations?(a) (x 1 , y 1 ) + (x 2 , y 2 ) = (x 1 + x 2 , y 1 + y 2 ) and r · (x, y) = (rx, y)(b) (x 1 , y 1 ) + (x 2 , y 2 ) = (x 1 + x 2 , y 1 + y 2 ) and r · (x, y) = (rx, 0)