Let the solution space U of the system of equations 2x1 + x2 + 3x3 − x4 + x5 = 0, 3x1 + 2x2 − 2x4 + x5 = 0, 3x1 + x2 + 9x3 − x4 + x5 = 0, be the subspace of R 5 endowed with the usual dot product. Find a basis for U and a basis for the orthogonal complement U ⊥.

To find a basis for the solution space U, we first need to solve the system of equations. We can do this by setting up an augmented matrix and reducing it to row-echelon form.

The augmented matrix for the system is:

[ 2  1  3 -1  1 | 0 ]
[ 3  2  0 -2  1 | 0 ]
[ 3  1  9 -1  1 | 0 ]

After performing row operations to reduce this matrix to row-echelon form, we get:

[ 1  0  3  0  0 | 0 ]
[ 0  1 -3  0  0 | 0 ]
[ 0  0  0  1  1 | 0 ]

From this, we can see that the solution to the system is given by:

x1 = -3x3
x2 = 3x3
x4 = -x5

We can express this solution in parametric vector form as:

x = x3[-3, 3, 1, 0, 0] + x5[0, 0, 0, -1, 1]

So, a basis for the solution space U is given by the vectors:

B = {[-3, 3, 1, 0, 0], [0, 0, 0, -1, 1]}

To find a basis for the orthogonal complement U⊥, we need to find vectors that are orthogonal to every vector in U. Since U is a subspace of R5, and dim(U) = 2, we know that dim(U⊥) = 5 - 2 = 3.

We can see that the vectors [1, 0, 0, 0, 0], [0, 1, 0, 0, 0], and [0, 0, 0, 1, -1] are orthogonal to both basis vectors of U, so a basis for U⊥ is given by:

B⊥ = {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 0, 1, -1]}