Final answer:
A basis for the subspace U is {(1,0,1,0), (0,1,0,1)} and its dimension is 2. These vectors are a linear combination that form every vector in U.
Step-by-step explanation:
To find a basis and the dimension of the subspace U defined by U= x₁ +x₂ =x₃ +x₄, we first look at the given equation that characterizes the subspace, rewrite it and express it in a more conventional vector form.
The given condition can be rewritten as:
x ₁ - x ₃ + x ₂ - x₄ = 0
By rearranging, we can express any vector in U as:
(x ₁,x ₂,x ₃ ,x₄) = x ₁(1,0,1,0) + x ₂(0,1,0,1)
This shows us that every vector in U can be expressed as a linear combination of the vectors (1,0,1,0) and (0,1,0,1).
Hence, a basis for U could be:
{(1,0,1,0), (0,1,0,1)}
To determine the dimension of U, we count the number of vectors in the basis. Since we have two vectors, the dimension of U is 2.
This aligns with the properties of vector addition in higher dimensions where vectors can be added in any order, which is commutative (A + B = B + A).