Final answer:
To determine the values of h that make the vectors span R3, we need to check for linear independence, find the null space, determine the spanning set, and find the eigenvalues.
Step-by-step explanation:
In order for the vectors a1, a2, and a3 to span R3, they must be linearly independent. This means that no vector can be written as a linear combination of the other vectors.
To check for linear independence, we can form an augmented matrix using the vectors as the columns and row-reduce it. If the row-reduced echelon form has no row of zeros, then the vectors are linearly independent.
To find the null space, we can solve the system of equations Ax = 0, where A is the matrix formed by the vectors a1, a2, and a3.
The spanning set of the vectors is the set of all possible linear combinations of the vectors. This can be represented as a subspace of R3.
The eigenvalues of the vectors can be found by solving the characteristic equation det(A - λI) = 0, where A is the matrix formed by the vectors a1, a2, and a3, λ is the eigenvalue, and I is the identity matrix.