Final answer:
To find a linearly independent set of vectors that spans the same R³ subspace as the given vectors, perform row reduction on the matrix formed by these vectors to identify the independent vectors.
Step-by-step explanation:
To determine a linearly independent set of vectors that spans the same subspace of R³ as the given vectors [2, 2, 0], [3, −3, 3], and [2, 0, 1], we can use the method of finding the basis of the column space of a matrix formed by these vectors.
Step 1: Form the Matrix from Vectors
First, arrange the given vectors as columns of a matrix:
A =
[2 3 2]
[2 −3 0]
[0 3 1]
Step 2: Perform Row Reduction
Next, we perform row reduction to bring the matrix to its row echelon form or further to reduced row echelon form (RREF).
For instance, if we obtain the following RREF of matrix A:
[1 0 0]
[0 1 0]
[0 0 1]
This means that all three vectors are already linearly independent. However, if the RREF has a row of zeros, then we eliminate the corresponding original vector. The remaining vectors will form a linearly independent set that spans the same space.
Step 3: Identify the Linearly Independent Set
If, for example, the RREF is:
[1 0 −½]
[0 1 ¼]
[0 0 0]
We would then conclude that the first two vectors are linearly independent and span the same subspace as the original set. The third vector is a linear combination of the first two and can be eliminated.
The linearly independent set would then be [2, 2, 0] and [3, −3, 3]. This set is capable of spanning the same subspace as all three original vectors.