Final answer:
To verify if the given sets are bases for ℝ3, the vectors must be checked for linear independence and spanning by performing row reduction on a matrix formed by the vectors as columns. If results in an identity matrix after row reduction, the set is a basis; otherwise, it might be linearly independent but not a basis, or it may not span ℝ3.
Step-by-step explanation:
To determine if the sets [1 0 -2], [3 2 -4], [-3 -5 1] are bases for ℝ3, we must check if they are linearly independent and span ℝ3. A set of vectors forms a basis for a vector space if and only if they are both linearly independent and span the space. This can be done by setting up the vectors as columns in a matrix and performing row reduction to see if we end up with an identity matrix.
Let's form a matrix with these vectors as its columns:
A =
[ 1 3 -3 ]
[ 0 2 -5 ]
[-2 -4 1 ]
Now, we apply Gaussian elimination or row reduction to obtain the row-echelon form (REF) or reduced row-echelon form (RREF) of this matrix. If the REF or RREF is an identity matrix, then the original vectors are linearly independent and form a basis for ℝ3.
After row reduction, if we get:
- An identity matrix, the set is a basis.
- A matrix with at least one row of zeros, the set doesn't span ℝ3 and is not a basis.
- A matrix with non-zero rows, but not an identity matrix, the vectors are linearly independent but do not form a basis since they don't span ℝ3.