Subscribe $4.99/month

Un-lock Verified Step-by-Step Experts Answers.

Find the Source, Textbook, Solution Manual that you are looking for in 1 click.

Our Website is free to use.
To help us grow, you can support our team with a Small Tip.

All the data tables that you may search for.

Need Help? We got you covered.

For Arabic Users, find a teacher/tutor in your City or country in the Middle East.

Products

Find the Source, Textbook, Solution Manual that you are looking for in 1 click.

For Arabic Users, find a teacher/tutor in your City or country in the Middle East.

Need Help? We got you covered.

Chapter 6

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