91Ó°ÊÓ

Find the projection matrix \(P\) onto the space spanned by \(a_{1}=(1,0,1)\) and \(a_{2}=\) \((1,1,-1)\)

Find a vector \(x\) orthogonal to the row space of \(A\), and a vector \(y\) orthogonat to the column space, and a vector \(z\) orthogonal to the nullspace: $$ A=\left[\begin{array}{lll} 1 & 2 & 1 \\ 2 & 4 & 3 \\ 3 & 6 & 4 \end{array}\right] $$

By choosing the correct vector \(b\) in the Schwarz inequality, prove that $$ \left(a_{1}+\cdots+a_{n}\right)^{2} \leq n\left(a_{1}^{2}+\cdots+a_{n}^{2}\right) $$ When does equality hold?

Show, by forming \(b^{\mathrm{T}} b\) directly, that Pythagoras's law holds for any combination \(b=x_{1} q_{1}+\cdots+x_{n} q_{n}\) of orthonormal vectors: \(\|b\|^{2}=x_{1}^{2}+\cdots+x_{n}^{2}\). In matrix terms, \(b=Q x\), so this again proves that lengths are preserved: \(\|Q x\|^{2}=\|x\|^{2}\).

If \(\mathbf{V}\) and \(\mathbf{W}\) are orthogonal subspaces, show that the only vector they have in common is the zero vector: \(\mathbf{V} \cap \mathbf{W}=\\{0\\}\).

If \(P\) is the projection matrix onto a \(k\)-dimensional subspace \(\mathbf{S}\) of the whole space \(\mathbf{R}^{n}\), what is the column space of \(P\) and what is its rank?

Project the vector \(b=(1,2)\) onto two vectors that are not orthogonal, \(a_{1}=(1,0)\) and \(a_{2}=(1,1)\). Show that, unlike the orthogonal case, the sum of the two onedimensional projections does not equal \(b\).

If the vectors \(q_{1}, q_{2}, q_{3}\) are orthonormal, what combination of \(q_{1}\) and \(q_{2}\) is closest to \(q_{3}\) ?

Find the orthogonal complement of the plane spanned by the vectors \((1,1,2)\) and \((1,2,3)\), by taking these to be the rows of \(A\) and solving \(A x=0\). Remember that the complement is a whole line.

(a) If \(P=P^{\mathrm{T}} P\), show that \(P\) is a projection matrix. (b) What subspace does the matrix \(P=0\) project onto?