Find a basis and the dimension of the subspace of generated by
{(1, -4, -2, 1), (1, -3, -1, 2), (3, -8, -2, 7)}.
To find a basis for the subspace generated by the given vectors, we need to determine which of the vectors are linearly independent. We can do this by forming a matrix with the given vectors as its columns and performing row reduction to determine if any rows of zeros are produced.
The matrix formed with the given vectors as its columns is:
[1 1 3]
[-4 -3 -8]
[-2 -1 -2]
[1 2 7]
Row reducing this matrix gives:
[1 0 1]
[0 1 2]
[0 0 0]
[0 0 0]
From this row reduced form, we can see that the first and second columns form a basis for the subspace. Therefore, a basis for the subspace is {(1, -4, -2, 1), (1, -3, -1, 2)}, and the dimension of the subspace is 2.
To find a basis and the dimension of the subspace generated by a set of vectors, we can follow these steps:
Step 1: Set up the augmented matrix with the given vectors as columns. In this case, the augmented matrix would look like this:
[1 1 3]
[-4 -3 -8]
[-2 -1 -2]
[1 2 7]
Step 2: Perform row operations to row-reduce the augmented matrix into reduced row echelon form.
After performing the row operations, we get the following reduced row echelon form:
[1 0 1]
[0 1 -2]
[0 0 0]
[0 0 0]
Step 3: Take the vectors corresponding to the non-zero rows in the reduced row echelon form as the basis for the subspace. In this case, the first two vectors form a basis:
{(1, 0, 1), (0, 1, -2)}
Step 4: Determine the dimension of the subspace by counting the number of vectors in the basis. In this case, the dimension of the subspace is 2.
To summarize, a basis for the subspace generated by {(1, -4, -2, 1), (1, -3, -1, 2), (3, -8, -2, 7)} is {(1, 0, 1), (0, 1, -2)}, and the dimension of the subspace is 2.