Question

For each of the following matrices, determine a basis for each of the subspaces $R\left(A^{T}\right), N(A), R(A),$ and $N\left(A^{T}\right):$ (a) $A=\left(\begin{array}{ll}{3} & {4} \\ {6} & {8}\end{array}\right)$ (b) $A=\left(\begin{array}{lll}{1} & {3} & {1} \\ {2} & {4} & {0}\end{array}\right)$ (c) $A=\left[\begin{array}{rr}{4} & {-2} \\ {1} & {3} \\ {2} & {1} \\ {3} & {4}\end{array}\right]$ (d) $A=\left(\begin{array}{llll}{1} & {0} & {0} & {0} \\ {0} & {1} & {1} & {1} \\ {0} & {0} & {1} & {1} \\ {1} & {1} & {2} & {2}\end{array}\right)$

          For each of the following matrices, determine a basis for each of the subspaces $R\left(A^{T}\right), N(A), R(A),$ and $N\left(A^{T}\right):$
(a) $A=\left(\begin{array}{ll}{3} & {4} \\ {6} & {8}\end{array}\right)$
(b) $A=\left(\begin{array}{lll}{1} & {3} & {1} \\ {2} & {4} & {0}\end{array}\right)$
(c) $A=\left[\begin{array}{rr}{4} & {-2} \\ {1} & {3} \\ {2} & {1} \\ {3} & {4}\end{array}\right]$
(d) $A=\left(\begin{array}{llll}{1} & {0} & {0} & {0} \\ {0} & {1} & {1} & {1} \\ {0} & {0} & {1} & {1} \\ {1} & {1} & {2} & {2}\end{array}\right)$

Added by James M.

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