Final answer:
The vector [1 5 7] cannot be written as a linear combination of [0 -1 -1], [0 0 -1], [-1 -1 0] because the given vectors cannot be scaled to produce the required positive integer in the first coordinate of [1 5 7].
Step-by-step explanation:
Writing the vector [1 5 7] as a linear combination of the vectors [0 -1 -1], [0 0 -1], [-1 -1 0] involves finding scalar multiples of these vectors that, when added, result in [1 5 7]. Let's denote the scalars as a, b, and c, respectively, and form the following equations:
- 0a + 0b - 1c = 1
- -1a + 0b - 1c = 5
- -1a - 1b + 0c = 7
Upon solving this system of equations, we'll find that no such scalars exist because the first equation implies that the vector contributions on the right-hand side would need to have a 1 in the first coordinate, which is impossible given the vectors we have. Since there are no vector contributions from the first two vectors in the first coordinate and the third vector has a negative contribution, it's impossible to achieve a positive integer like 1 as required for the first coordinate of [1 5 7]. Therefore, the vector [1 5 7] cannot be written as a linear combination of [0 -1 -1], [0 0 -1], [-1 -1 0].
Remember that vector addition is both associative and commutative, meaning that the order in which vectors are added does not affect the sum. However, this property does not help in this case because the vectors provided are not capable of combining to form [1 5 7] regardless of the order in which they are added.