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Diffstat (limited to 'li/hw6.tex')
| -rw-r--r-- | li/hw6.tex | 14 |
1 files changed, 9 insertions, 5 deletions
@@ -12,8 +12,12 @@ $$P = A(A^TA)^{-1}A^T = \bmatrix{1&1\cr 1&-1\cr -2&4} \fr1{22}\bmatrix{9&4\cr4&3}\bmatrix{1&1&-2\cr1&-1&4} = \fr1{22}\bmatrix{20&6&2\cr 6&4&-6\cr2&-6&20}.$$ -$$Pb = \fr1{22}\bmatrix{20&6&2\cr 6&4&-6\cr2&-6&20}\bmatrix{1\cr2\cr7} -= \fr1{22}\bmatrix{46\cr-28\cr130}$$ +$$p = Pb = \fr1{22}\bmatrix{20&6&2\cr +6&4&-6\cr2&-6&20}\bmatrix{1\cr2\cr7} = +\fr1{22}\bmatrix{46\cr-28\cr130}$$ +With $b = p + q,$ and $p$ known, $q = b-p,$ and $q$ is in the left null +space of $A$ by the definition of orthogonality with the column space +which contains $p.$ \noindent{\bf 7.} @@ -47,12 +51,12 @@ longer distance. \noindent{\bf 16.} -A is a $3\times2$ matrix, $Q$ is also a $3\times2$ matrix, and $R$ is a -$2\times2$ matrix. - $$A = \bmatrix{1&1\cr2&3\cr2&1} = \bmatrix{1/3&0\cr 2/3&1/\sqrt2\cr2/3&-1/\sqrt2}\bmatrix{3&3\cr0&\sqrt2} = QR.$$ +Generally, $A$ is an $m\times n$ matrix, $Q$ is an $m\times r$ (where +$r$ is the rank of $A$) matrix, and $R$ is an $r\times r$ matrix. + \noindent{\bf 17.} $$Pb = QQ^Tb = \bmatrix{1/9&2/9&2/9\cr 2/9&17/18&-1/18\cr |