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.
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.
To find the possible dimensions for V, we need to consider the number of linearly independent vectors that span the subspace.
In R^3, the maximum number of linearly independent vectors that can span a subspace is 3. Therefore, the possible dimensions for V are 0, 1, 2, or 3.
Now, let's prove that if V is a subspace of R^3, either V = {0}, or V is a straight line passing through the origin, or V is a plane passing through the origin.
Case 1: V = {0}
If V only contains the zero vector, then V = {0}.
Case 2: V is a straight line passing through the origin
If V contains a single nonzero vector, say v, then V will be a straight line passing through the origin since all scalar multiples of v will also belong to V.
Case 3: V is a plane passing through the origin
If V contains two linearly independent vectors, say v1 and v2, then all linear combinations of v1 and v2 will form a plane passing through the origin. This is because any vector in the plane can be expressed as a linear combination of v1 and v2.
Therefore, if V is a subspace of R^3, it can have a dimension of 0, 1, 2, or 3, and it will be either {0}, a straight line passing through the origin, or a plane passing through the origin.