Let {(3,5,-1);(7,2,-2)} be a spanning set. Determine the space generated by S.

To determine the space generated by the given spanning set, we need to find the span of the vectors (3, 5, -1) and (7, 2, -2). The span of a set of vectors is the set of all possible linear combinations of those vectors. In other words, it is the set of all vectors that can be obtained by multiplying each vector in the set by a scalar and adding them together. So, let's find the span of these two vectors: Let's assume the span of the given set is denoted by S: S = Span{(3, 5, -1), (7, 2, -2)} To find the span, we need to find all possible linear combinations of these vectors. Let's take a generic vector (a, b, c) that belongs to the span S. $$(a, b, c) = x(3, 5, -1) + y(7, 2, -2)$$ Where x and y are scalars. Expanding: $$(a, b, c) = (3x, 5x, -x) + (7y, 2y, -2y) = (3x + 7y, 5x + 2y, -x - 2y)$$ So, any vector (a, b, c) in the span S can be written as (3x + 7y, 5x + 2y, -x - 2y) for some values of x and y. Therefore, the space generated by S is the set of all vectors of the form (3x + 7y, 5x + 2y, -x - 2y) where x and y are real numbers. In other words, the space generated by S is the set of all vectors in R^3 that can be obtained by multiplying (3, 5, -1) and (7, 2, -2) by scalars and adding them together.