4.4.29. For each of the following matrices A, (i) find a basis for each of the four fundamental subspaces; (ii) verify that the image and cokernel are orthogonal complements; (iii) verify that the coimage and kernel are orthogonal complements: (a) egin{pmatrix} 1 & -2 \ 2 & -4 end{pmatrix}, (b) egin{pmatrix} 5 & 0 \ 1 & 2 \ 0 & 2 end{pmatrix}, (c) egin{pmatrix} 0 & 1 & 2 \ -1 & 0 & -3 \ -2 & 3 & 0 end{pmatrix}, (d) egin{pmatrix} 1 & 2 & 0 & 1 \ -1 & 1 & 3 & 1 \ 0 & 3 & 3 & 2 end{pmatrix}
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For this solution, we will use matrix (a) as an example: \[ A = \begin{pmatrix} 1 & -2 \\ 2 & -4 \end{pmatrix} \] Show moreā¦
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