The possible dimensions for V are 0, 1, or 2.
To show that if V ⊆ ℝ^3, then either V = {0}, or V is a straight line passing through the origin, or V is a plane passing through the origin, we can consider the dimensions of V.
If dim(V) = 0, then V = {0}. This is a subspace consisting only of the zero vector.
If dim(V) = 1, then V is a straight line passing through the origin. This means that V can be represented as the span of a single vector v = (a,b,c) ≠ 0 in V. Any scalar multiple of v will lie on this line.
If dim(V) = 2, then V is a plane passing through the origin. This means that V can be represented as the span of two linearly independent vectors v1 = (a1,b1,c1) and v2 = (a2,b2,c2) in V. Any linear combination of v1 and v2 will lie on this plane.
In summary, if V is a subspace of ℝ^3, then its possible dimensions are 0, 1, or 2. If dim(V) = 0, V = {0}. If dim(V) = 1, V is a straight line passing through the origin. If dim(V) = 2, V is a plane passing through the origin.
Let V be a subspace of (3. What are the possible dimensions for V? Show that if
V ((3, then either V = {0}, or V is a straight line passing through the origin, or V is a plane passing through the origin.
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