Abstract Algebra: Theory and Applications · Problem 20.7

Fields — Thomas Judson

Mathematics — Abstract Algebra · subspaces-basis-dimension

Problem

Which of the following sets are subspaces of R3\mathbb{R}^3? If the set is indeed a subspace, find a basis for the subspace and compute its dimension. (a) {(x1,x2,x3):3x1−2x2+x3=0}\{(x_1, x_2, x_3) : 3x_1 - 2x_2 + x_3 = 0\} (b) {(x1,x2,x3):3x1+4x3=0, 2x1−x2+x3=0}\{(x_1, x_2, x_3) : 3x_1 + 4x_3 = 0,\, 2x_1 - x_2 + x_3 = 0\} (c) {(x1,x2,x3):x1−2x2+2x3=2}\{(x_1, x_2, x_3) : x_1 - 2x_2 + 2x_3 = 2\} (d) {(x1,x2,x3):3x1−2x22=0}\{(x_1, x_2, x_3) : 3x_1 - 2x_2^2 = 0\} Hint. (a) Subspace of dimension 2 with basis {(1,0,−3),(0,1,2)}\{(1, 0, -3), (0, 1, 2)\}; (d) not a subspace

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