Determine Whether The Set Is A Subspace
Determine whether or not the set given is a subspace in the first six problems
Evaluate each of the following sets in R3 to determine whether they are subspaces. Justify your answers with clear reasoning, including checks for closure under addition and scalar multiplication, as well as the presence of the zero vector if appropriate.
Paper For Above instruction
In linear algebra, a subset W of a vector space V (here, R3) is a subspace if it satisfies three criteria: it contains the zero vector, it is closed under vector addition, and it is closed under scalar multiplication. Analyzing each problem involves checking these properties for each set defined by specific conditions.
Problem 1: Set W = {x ∈ R3 | x₂2 = x₂ + x₃}
The set W is defined by the quadratic relation x₂2 = x₂ + x₃. To determine if W is a subspace, first check if the zero vector (0,0,0) is in W:
Substitute: 02 = 0 + 0 → 0 = 0, which holds true, so the zero vector is in W.
Next, test for closure under addition and scalar multiplication. Consider two arbitrary vectors x = (x₁, x₂, x₃) and y = (y₁, y₂, y₃) in W, satisfying the defining equation. Their sum x + y should also satisfy the relation; however, because x₂2 = x₂ + x₃ and y₂2 = y₂ + y₃, the sum's second component is the sum of the individual second components, but (x₂ + y₂)2 ≠ x₂2 + y₂2. Furthermore, this quadratic relation will not generally be preserved under addition or scalar multiplication unless specific conditions are met. For example, adding vectors in W does not necessarily produce a vector satisfying the same relation, which violates closure.
Therefore, W is not closed under addition, and thus, it is not a subspace.
Problem 2: Set W = {x ∈ R3 | x₁ = 3x₃, x₂ = –4x₃}
This set describes vectors where x₁ and x₂ are directly proportional to x₃. The set W can be expressed as:
W = { (3x₃, –4x₃, x₃) | x₃ ∈ R } = x₃ (3, –4, 1)
This is the span of the single vector (3, –4, 1). Since spans of vectors are subspaces, W is a line through the origin.
It contains the zero vector when x₃ = 0, is closed under addition (since any linear combination of vectors in W remains in W), and is closed under scalar multiplication.
Therefore, W is indeed a subspace of R3.
Problem 3: Set W = {x ∈ R3 | x₁ x₂ = x₃}
Here, W consists of all vectors satisfying a quadratic relation. The zero vector (0, 0, 0) satisfies 0×0=0, so it is in W.
However, testing closure under addition reveals problems: Consider x = (1, 0, 0) and y = (0, 1, 0). Both satisfy x₁ x₂ = x₃, as 1×0=0 and 0×1=0. Sum: x + y = (1, 1, 0), which gives 1×1=1 ≠ 0, so x + y is not in W.
This example shows W is not closed under addition.
Thus, W is not a subspace.
Problem 4: Set W = {x ∈ R3 | xT a = xT b} with vectors a, b fixed
The set W consists of all vectors x satisfying the linear equation:
xT a = xT b, or equivalently, xT (a – b) = 0.
Such a set represents all vectors orthogonal to the fixed vector (a – b), which forms a hyperplane passing through the origin.
This set contains the zero vector because 0T (a – b) = 0. Closure under addition and scalar multiplication hold due to the linear nature of the defining equation.
Thus, W is a subspace of R3.
Problem 5: Set W = {x ∈ R3 | x₁ ≥ x₂}
This set is defined by an inequality involving the components of vectors. Since inequalities do not define a linear subspace (they do not satisfy closure under scalar multiplication, particularly for negative scalars), W cannot be a subspace.
For example, take x = (1, 0, 0) in W: 1 ≥ 0. Multiply by –1: (–1, 0, 0), which has component 1 ≤ 0, so it does not satisfy the inequality — not in W.
Therefore, W is not a subspace.
Problem 6: Set W = {x ∈ R3 | A x = v for a fixed matrix A and vector v}
This set is the solution set to a linear system with a fixed right-hand side v. For W to be a subspace, the set of solutions must contain 0 and be closed under addition and scalar multiplication.
If v ≠ 0, the set W does not contain the zero vector (since A 0 ≠ v), and thus it cannot be a subspace.
If v = 0, then W = Null(A), the null space of A, which is a subspace.
Hence, W is a subspace iff v = 0.
Problems 7-10: Assessing subspaces formed by unions, intersections, sums, and inequalities
Problem 7: W = U ∪ V, where U and V are subspaces of R3
The union of two subspaces U and V is generally not a subspace unless one is contained in the other. Since the union may not be closed under addition, W typically fails to be a subspace.
Problem 8: W = U ∩ V, where U and V are subspaces
The intersection of subspaces is always a subspace, as it contains the zero vector, is closed under addition and scalar multiplication.
Problem 9: W = U + V, the sum of subspaces
The sum of subspaces is also a subspace, as it is the set of all sums u + v, with u ∈ U, v ∈ V, and is closed under addition and scalars.
Problem 10: W = {x ∈ R3 | x₁2 + x₂2 + x₃2 ≥ 0}
This set includes all vectors in R3 since the sum of squares is always non-negative. It contains the zero vector, but it also includes vectors where the sum is strictly greater than zero.
However, it is not closed under scalar multiplication because multiplying a vector in W by –1 will preserve the sum of squares being non-negative. But to be a subspace, it must be closed under multiplication by all scalars, and the zero vector is included, but closure under scalar multiplication involving negative scalars for the entire set is maintained.
Thus, W is a subspace—a cone containing the zero vector.
References
- Lay, D. C., Lay, S. R., & McDonald, J. J. (2016). Linear algebra and its applications (5th ed.). Pearson.
- Strang, G. (2016). Introduction to linear algebra (5th ed.). Wellesley-Cambridge Press.
- Anton, H., & Rorres, C. (2013). Elementary linear algebra (11th ed.). Wiley.
- Leon, S. J. (2007). Linear algebra with applications (8th ed.). Pearson.
- Hoffman, K., & Kunze, R. (1971). Linear algebra (2nd ed.). Prentice-Hall.
- Larson, R., & Edwards, B. (2013). Elementary linear algebra (6th ed.). Brooks Cole.
- Fitzpatrick, P. (2012). Linear algebra. American Mathematical Society.
- Hefferon, J. (2017). Linear algebra. Springer.
- Halmos, P. R. (1974). Finite-dimensional vector spaces. Springer.
- Kolman, B., & Hall, R. (2018). College algebra and trigonometry (9th ed.). Pearson.