Question

Find orthonormal bases for the four fundamental subspaces associated with the following matrices:

a.(1 -1)
(-3 3)
b.(-1 0 2)
(1 1 -1)
(0 1 1)
c.(1 0 1 o)
(1 1 1 1)
(-1 2 0 1)
d.(1 2 1)
(0 -2 1)
(-1 0 -2)
(1 -2 3)

Answer 1

An orthonormal basis for the four fundamental subspaces of a matrix is found by identifying the column, row, null, and left null spaces and applying the Gram-Schmidt process. The specific bases depend on the given matrix, which is not provided here. The general approach is systematic for any matrix.

Step-by-step explanation:

Finding orthonormal bases for the four fundamental subspaces associated with a matrix involves identifying the column space, row space, nullspace, and left nullspace of the matrix, and then using the Gram-Schmidt process to orthonormalize each subspace. To provide a meaningful answer, we need the actual matrices in question, which are not provided in the snippet. Once the matrices are available, one can perform the necessary operations (such as row reductions for the Row Reduced Echelon Form (RREF), finding eigenvectors, etc.) to extract the bases for the respective subspaces and then apply the Gram-Schmidt process to orthonormalize those bases.

The general approach is systematic and applies well to any matrix. The column space basis can be found from the pivot columns of the matrix, the row space basis from the pivot rows, the nullspace from solving the system when it's equal to the zero vector, and the left nullspace from the rows of the nullspace of the matrix's transpose.