Question

Suppose that v1 = (2, 1, 0, 3), v2 = (3, -1, 5, 2), and v3 = (-1, 0, 2, 1). Which of the following vectors are in span {v1, v2, v3}?

(a) (2, 3, -7, 3)
(b) (0, 0, 0, 0)
(c) (1, 1, 1, 1),
(d) (-4, 6, -13, 4)

Answer 1

Only the zero vector (0, 0, 0, 0) is guaranteed to be in the span of any set of vectors, including {v1, v2, v3}. To check if the other given vectors are in the span, one would need to solve a system of linear equations for each vector.

Step-by-step explanation:

The question deals with determining which vectors are in the span of three given vectors v1, v2, and v3. To find out whether a vector is in the span of these vectors, we must check if there exists a combination of scalar multiples of v1, v2, and v3 that can produce the given vector.

Let's analyze the provided vectors (a) (2, 3, -7, 3), (b) (0, 0, 0, 0), (c) (1, 1, 1, 1), and (d) (-4, 6, -13, 4) to see which are in the span of {v1, v2, v3}:

• (b) The zero vector (0, 0, 0, 0) is always in the span of any set of vectors because it can be represented as a linear combination of the vectors with all scalar coefficients being zero.
• (c) To check if vectors (a), (c), and (d) are in the span of {v1, v2, v3}, we would set up a system of linear equations with the scalars as unknowns and attempt to solve. If there is a solution, the vector is in the span.

Without further calculation provided, we can only definitively say that the zero vector (b) is in the span of {v1, v2, v3}. For vectors (a), (c), and (d), we would need to perform the actual calculations to determine whether they are in the span or not.