Question

W={(x₁, x₂, x₃) ∈ ℝ³: x₁-x₃=0}

(a) What is a basis for W?
(b) Let B be the set you found in part (a). Consider the subspace U

Answer 1

To find a basis for the set W, rewrite the equation as x₁ = x₃ and select a vector that satisfies it, such as (1, 0, 1).

Step-by-step explanation:

In this case, W is defined as the set of all vectors (x₁, x₂, x₃) in ℝ³ that satisfy the equation x₁ - x₃ = 0. To find a basis for W, we need to determine a set of linearly independent vectors that span W.

Note that the equation x₁ - x₃ = 0 can be rewritten as x₁ = x₃. So, any vector of the form (x₃, x₂, x₃) will satisfy the equation and be an element of W.

A basis for W can then be given by a single vector, such as (1, 0, 1). This vector is linearly independent, as there is no non-trivial solution to the equation k(1, 0, 1) = (0, 0, 0).