Final answer:
To find a vector b that is not a linear combination of a1=⟨1,0,1⟩,a2=⟨1,1,0⟩, and a3=⟨0,−1,1⟩, you need to find a vector that is not in the span of these three vectors.
Step-by-step explanation:
To find a vector b that is not a linear combination of a1 = ⟨1,0,1⟩, a2 = ⟨1,1,0⟩, and a3 = ⟨0,−1,1⟩, we need to find a vector that is not in the span of these three vectors. The span of a set of vectors is the set of all possible linear combinations of those vectors.
We can use the fact that if a vector is in the span of a set of vectors, its coordinates can be expressed as a linear combination of the coordinates of those vectors. Therefore, we need to find a vector b whose coordinates cannot be expressed as a linear combination of the coordinates of a1, a2, and a3.
One way to do this is to find a vector b that is linearly independent of a1, a2, and a3. For example, we can choose b = ⟨1,1,1⟩. Since the coordinates of b cannot be expressed as a linear combination of the coordinates of a1, a2, and a3, b is not a linear combination of a1, a2, and a3.