Question

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unena11Ic.doc 10. Determine whether the given collection is linearly independent in P3. A) x3-x, x2- 1, x, 1) B) 1, 1+x, x + x2, x2+ x}e8onil A C) {x3 +x2+x+ 1, x3 + x2 + x, x3 + x2, x3) D) {x3 +x2 +x +1, x+x, x +1}woor aan 11. Determine whether the given collection is a spanning set of R. A) <0, 1, 1>, <1, 0, 1>, <1, 1, 0>} B) <0, 0, 1>, <0, 1, 1>, <l, 1, 1>}oon aonn C) <0, 0, 1>, <0, 1, 1>, <1, 1, 0> D) {<0, 0, 1>, <0, 1, 0>, <0, 1, 1>} E) <0, 0, 1, 2>, <0, 1, 2, 0>, <1, 2, 0, 0>, <1, 2, 1, 2>} F) <3, 2, 1>,<, 1, 2, 3> H 52 91 bb Toueu odh ebsco

Answer 1

(A) c_1 1 + c_2 x + c_3 (x z_1) + c_4 (x^3 - x) = 0 Rightarrow (c_1 - c_3) + x (c_2 - c_4) + x^2 (c_3) + c_4 x^3 = 0 Rightarrow c_3 = c_4 = 0 c_1 - c_3 = 0, c_2 - c_4 = 0 Rightarrow c_1 = c_2 = c_3 = c_4 = 0 So, (A) is independent (B) c_1 1 + c_2 (1 + x) + c_3 (x + x^2) + c_4 (x^2 + x^3) = 0 Rightarrow (c_1 + c_2) + x (c_2 + c_3) + x^2 (c_3 + c_4) + c_4 x^3 = 0 Rightarrow c_1 + c_2 = 0, c_2 + c_3 = 0, c_3 + c_4 = 0, c_4 = 0 Rightarrow c_4 = 0 = c_3 = c_2 = c_1 So, (B) is linearly independent (c) c_1 (x^3) + c_2 (x^3 + x^2) + c_3 (x + x^2 + x^3) + c_4 (1 + x + x^2 + x^3) = 0 Rightarrow c_4 + x (c_3 + c_4) + x^2 (c_2 + c_3 + c_4) + x^3 (c_1 + c_2 + c_3 + c_4) = 0 Rightarrow c_4 = 0, c_3 + c_4 = 0, c_2 + c_3 + c_4 = 0, c_1 + c_2 + c_3 + c_4 = 0 Rightarrow c_4 = c_3 = c_2 = c_1 = 0