Question

Let S be the the following subspace of R4: S=\operatorname{span}\left\{\left[\begin{array}{l}

1 \\

0 \\

0 \\

1

\end{array}\right],\left[\begin{array}{l}

1 \\

1 \\

1 \\

1

\end{array}\right]\right\} ■) Give dim(S) and dim(S¹). S is the orthogonal complement of S. (b) Compute Ps and Ps¹, the projection matrices on S and St, respectively. (c) What are the relationships between the column spaces and nullspaces of Ps and Ps+,i.e., C(Ps), N(Ps), C(Ps+), and N(Ps+). (d) Give orthonormal bases for each of C'(Ps), N(Ps), C(Ps±), and N(Ps+). Give the eigenvalues of Ps, along with their algebraic and geometric multiplicities.

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