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## Q. 6.2.11

Summarizing Several Results

Consider the linear transformation $T:R^n → R^m$ represented by $T$(x) = Ax. Find the nullity and rank of $T,$ and determine whether $T$ is one-to-one, onto, or neither.

a. $A =\left [ \begin{matrix} 1 & 2 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} \right ]$                       b. $A =\left [ \begin{matrix} 1 & 2 \\ 0 & 1 \\ 0 & 0 \end{matrix} \right ]$

c. $A = \left [ \begin{matrix} 1 & 2 & 0 \\ 0 & 1 & -1 \end{matrix} \right ]$                    d. $A = \left [ \begin{matrix} 1 & 2 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{matrix} \right ]$

## Verified Solution

Note that each matrix is already in row-echelon form, so its rank can be determined by inspection.

 T:$R^n → R^m$ Dim(domain) Dim(range) Rank(T) Dim(kernel)  Nullity(T) One-to-One Onto a. T:R³ → R³ 3 3 0 Yes Yes b. T:R² → R³ 2 2 0 Yes No c. T:R³ → R² 3 2 1 No Yes d. T:R³ → R³ 3 2 1 No No