For each of the following subspaces of F3, determine whether it is a subspace of F3. Recall that x ∈ Fn if and only if: (a) {x ∈ F3 | x1 + 2x2 + 3x3 = 0} (b) {x ∈ F3 | x1 + 2x + 3x3 = 4} (c) {x ∈ F3 | x1 + x2 + x3 = 0} (d) {x ∈ F3 | xC = 5x} (e) {a + 3b}
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Closure under addition: Let's take two vectors, u and v, that satisfy the equation x1 + 2x2 + 3x3 = 0. If we add them together, we get a new vector w where w1 = u1 + v1, w2 = u2 + v2, and w3 = u3 + v3. We can see that w also satisfies the equation x1 + 2x2 + 3x3 Show more…
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