# Finding standard matrix for T

• Jul 31st 2008, 09:51 AM
Finding standard matrix for T
This problem was on one of my tests, but I can't find any similar problems in the book or in HW to help me figure it out, and I'm crap at using the book to help solve problems

Let t : R2 -> R3 be defined as T(x) = [x1,x1+x2,0] , also let b = [1,1,1]

A. Find the standard matrix for T
B. Is T a linear transformation?
C. Is b in the range of T?
D. Is T onto, or one-to-one?
• Jul 31st 2008, 11:30 AM
CaptainBlack
Quote:

This problem was on one of my tests, but I can't find any similar problems in the book or in HW to help me figure it out, and I'm crap at using the book to help solve problems

Let t : R2 -> R3 be defined as T(x) = [x1,x1+x2,0] , also let b = [1,1,1]

A. Find the standard matrix for T
B. Is T a linear transformation?
C. Is b in the range of T?
D. Is T onto, or one-to-one?

A. Now there are two conventions for the definition of the matrix for T. The convention I am used to the vectors are column vectors, and we write the action as Tx, the other treats the vectors as row vectors and has action xT.

In the first case the columns are the images of the natural basis of the domain

$
{\bf T} = \left[ {\begin{array}{*{20}c}
1 & 0 \\
1 & 1 \\
0 & 0 \\
\end{array}} \right]
$

In the second case the rows are the images of the natural basis:

$
{\bf T} = \left[ {\begin{array}{*{20}c}
1 & 1 & 0 \\
0 & 1 & 0 \\
\end{array}} \right]
$

B. You check this

C. The last component of $\bold{T}(\bold{x})$ for all $\bold{x} \in \mathbb{R}^2$ is zero so is $\bold{b}$ of this form?

D. Look at the answer to C. to see if $\bold{T}$ is onto. To see if it is one-to-one consider if there are distinct $\bold{x}$ and $\bold{y} \in \mathbb{R}^2$ such that:

$\bold{T}(\bold{x})=\bold{T}(\bold{y}).$

RonL