for each of the following subsets of F^3, determing whether it is a subspace of
ID: 2940686 • Letter: F
Question
for each of the following subsets of F^3, determing whether it is a subspace of F^3:a) {(x1, x2, x3) element of F^3: x1 + 2x2 + 3x3 = 0} b) {(x1, x2, x3) element of F^3: x1 + 2x2 + 3x3 = 4} c) {(x1, x2, x3) element of F^3: x1x2x3 = 0} d) {(x1, x2, x3) element of F^3: x1 = 5x3} for each of the following subsets of F^3, determing whether it is a subspace of F^3:
a) {(x1, x2, x3) element of F^3: x1 + 2x2 + 3x3 = 0} b) {(x1, x2, x3) element of F^3: x1 + 2x2 + 3x3 = 4} c) {(x1, x2, x3) element of F^3: x1x2x3 = 0} d) {(x1, x2, x3) element of F^3: x1 = 5x3}
Explanation / Answer
(a) is a plane through origin and is a subspace. (b) is a plane passing away from origin and so is a subspace. (c) is the trivial subspace while x1x2x3=0 , the only possibility is x1=x2=x3=0 and so, this is the subspace is the zero space {0}. (d) is a line parallel to XZ plane comfertably. so, this is a subspace of dimension 1.
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