which of the following sets of vectors span R^3?
a.){(1, -1, 2), (0, 1, 1)}
b.) {1, 2, -1), (6, ,3, 0), (4, -1, 2), (2, -5, 4)}
c.) {(2, 2, 3), (-1, -2, 1), (0, 1, 0)}
d.) {(1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 1)}
can someone show the steps to check for one of them and i will try to do the rest.
Put the vectors as rows or columns in a matrix and do Gaussian Elimination (note that row rank = column rank).
You don't have to do this for a) because you can't span R^3 with two vectors. Also, it is clear that the vectors listed in d) span R^3
In case of b) after Gaussian Elimination, you should find that the rank of the matrix is 2. This means that te four vectors span a two dimensional subspace of R^3 (the reduced matrix indicates exactly what subpace)
In case of c) you should find that the rank is 3, so the three listed vectors span R^3.
i need help
To check if a set of vectors spans R^3, we can use Gaussian Elimination. Let's take the set of vectors in option b.) as an example.
b.) {(1, 2, -1), (6, 3, 0), (4, -1, 2), (2, -5, 4)}
Step 1: Set up the matrix by taking the vectors as rows or columns. Let's use rows for this example:
| 1 2 -1 |
| 6 3 0 |
| 4 -1 2 |
| 2 -5 4 |
Step 2: Perform Gaussian Elimination to reduce the matrix to its row echelon form. This involves performing row operations to transform the matrix into an upper triangular form, and then further transforming it to its reduced row echelon form.
Applying Gaussian Elimination, we perform the following row operations:
Row 2 - 6 * Row 1:
| 1 2 -1 |
| 0 -9 6 |
| 4 -1 2 |
| 2 -5 4 |
Row 3 - 4 * Row 1:
| 1 2 -1 |
| 0 -9 6 |
| 0 -9 6 |
| 2 -5 4 |
Row 4 - 2 * Row 1:
| 1 2 -1 |
| 0 -9 6 |
| 0 -9 6 |
| 0 -9 6 |
Row 3 - Row 2:
| 1 2 -1 |
| 0 -9 6 |
| 0 0 0 |
| 0 -9 6 |
Row 4 - Row 2:
| 1 2 -1 |
| 0 -9 6 |
| 0 0 0 |
| 0 7 0 |
Step 3: Examine the reduced row echelon form of the matrix. The rank of the matrix is determined by the number of non-zero rows in the reduced row echelon form.
| 1 2 -1 |
| 0 -9 6 |
| 0 0 0 |
| 0 0 0 |
In this case, there are only two non-zero rows, and thus the rank of the matrix (and the set of vectors) is 2. This means that the four vectors in option b.) span a two-dimensional subspace of R^3, but not the entire R^3.
Now, you can follow the same steps to check the remaining options yourself. For option c.), you should find that the rank is 3, indicating that the three listed vectors span R^3.