## Chapter 6

## Q. 6.2.11

## 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 ]

## Step-by-Step

## 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 |