Determine whether {v1, v2, v3} is a basis for R3
v1 = [1]
[-3]
[1]
v2 = [-3]
[8]
[4]
v3 = [2]
[-2]
[2]
Put the vectors as rows or columns in a matrix and perform Gaussian reduction to determne the row or column rank. In this case you should find that the rank is 3, therefore te vectors span R^3
To determine whether {v1, v2, v3} is a basis for R3, we need to check if these vectors span R3 and are linearly independent.
1. We can put the vectors as rows or columns in a matrix A:
A = [v1 | v2 | v3]
= [1 -3 2]
[-3 8 -2]
[1 4 2]
2. Next, we perform Gaussian elimination to determine the row rank or column rank of matrix A. The row rank is equal to the column rank, so we can use either one.
Reducing the matrix A to row-echelon form:
R = [1 -3 2]
[0 1 1]
[0 0 0]
3. The row rank of matrix A is equal to the number of nonzero rows in the row-echelon form. In this case, there are only two nonzero rows.
Since the row rank (or column rank) is less than the number of columns (which is 3), the vectors {v1, v2, v3} do not span R3.
Therefore, {v1, v2, v3} is not a basis for R3.