Question

Find a non-trivial way, if possible, of expressing the 0 as a linear combination of the vectors in set S if S={(2,1−3),(1,2,−2),(1,−4,0),(−1,7,−1)}

Answer 1

To express the null vector as a linear combination of the vectors in set S, we must find a non-trivial solution to the system of linear equations derived from the vectors' components. This involves finding scalars that, when multiplied with the given vectors and summed up, result in the null vector.

Step-by-step explanation:

To express the null vector as a linear combination of the vectors in set S, we can write an equation as a sum of vectors in S, each multiplied by a scalar, that equals the null vector:

Let S = {(2,1,-3), (1,2,-2), (1,-4,0), (-1,7,-1)}

We need to find scalars a, b, c, and d such that:

a(2,1,-3) + b(1,2,-2) + c(1,-4,0) + d(-1,7,-1) = (0,0,0)

To solve this, we will set up a system of linear equations based on the components of the vectors:

1. 2a + b + c - d = 0
2. a + 2b - 4c + 7d = 0
3. -3a - 2b + d = 0

We can use methods such as substitution, elimination, or matrix operations (e.g., Gaussian elimination) to find a solution where a, b, c, and d are not all zero (a non-trivial solution). If possible, this would give us the required linear combination to express the null vector.