Question

Determine whether the set s spans r3. if the set does not span r3, then give a geometric description of the subspace that it does span. s = {(6, 8, 2), (−1, 2, 6), (2, −3, 4)}

Answer 1

The set S = {(6, 8, 2), (−1, 2, 6), (2, −3, 4)} spans R^3, as the vectors are linearly independent after performing row reduction on the matrix they form.

Step-by-step explanation:

To determine whether the set S = {(6, 8, 2), (−1, 2, 6), (2, −3, 4)} spans R^3, we need to verify if the vectors in set S are linearly independent and cover all of R^3. A step by step explanation goes as follows:

1. Construct a matrix with the vectors as rows.
2. Perform row reduction to find the row echelon form of the matrix.
3. Determine if there are any rows that consist entirely of zeros, which would indicate linear dependence.

If after row reduction the matrix has three non-zero rows, the vectors are linearly independent, and the set spans R^3. If the row reduction results in a row of zeros, the vectors do not span R^3, and they span a subspace of lower dimension. In this case, if the matrix has two non-zero rows, the set spans a plane in R^3, and if there is only one non-zero row, it spans a line.

Upon performing row reduction on the matrix formed by S, we find it results in three non-zero rows, indicating that the vectors in set S are linearly independent and span R^3. No subspace description is necessary.