# Question regarding linear transformation

1. May 9, 2005

### Mathman23

Hi

I have a linear transformation T which maps $$\mathbb{R}^3 \rightarrow \mathbb{R}^2$$ a A is the standard matrix for the linear transformation.

I'm suppose to determain that T maps $$\mathbb{R}^3 \rightarrow \mathbb{R}^2$$

I was told by my professor about the following theorem.

If $$T:\mathbb{R}^n \rightarrow \mathbb{R}^m$$ is a linear transformation and A is the standard matrix for T. Then

a/ T maps $$\mathbb{R}^n \rightarrow \mathbb{R}^m$$ if and only if A $$\mathrm{span} \{ \mathbb{R}^m \}$$

b/ T is one-to-one if and only if columns of A are linearly independent.

If I then apply (a) from to my problem:

$$A = \left[ \begin{array}{ccc} 1 & 0 & 4 \\ 2 & 1 & 6\\ \end{array} \right ]$$

A being the standard matrix of the linear transformation.

A can also be written:

$$\left[ \begin{array}{cc} 1 \\2 \end{array} \right ] \mathrm{x} + \left[ \begin{array}{cc} 0 \\1 \end{array} \right ] \mathrm{y} + \left[ \begin{array}{cc} 4 \\6 \end{array} \right ] \mathrm{z} = \left[ \begin{array}{cc} 0 \\0 \end{array} \right ]$$

This is equal to:

$$\nonumber x + 4z = 0$$

$$\nonumber 2x + y + 6z = 0$$

x in equation 1 can be written as $$x = -4z$$

If I insert that x into equation 2 then I get

$$-2z + y = 0$$

But what do I then do to prove point (a) in the theorem ???

Sincerley and many thanks in advance

Fred

2. May 11, 2005

### HallsofIvy

Staff Emeritus
Actually this: "I have a linear transformation T which maps $$\mathbb{R}^3 \rightarrow \mathbb{R}^2$$
a A is the standard matrix for the linear transformation.

I'm suppose to determain that T maps $$\mathbb{R}^3 \rightarrow \mathbb{R}^2$$" doesn't quite make sense! I assume you mean simply that T is from$$\mathbb{R}^3$$to $$\mathbb{R}^2$$ and you want to prove this is an "onto" mapping.
Okay, you are given A= $$A = \left[ \begin{array}{ccc} 1 & 0 & 4 \\ 2 & 1 & 6\\ \end{array} \right ]$$

I don't understand why you then set Ax= 0: that would prove that the columns are linearly independent which is your professor's (b), not (a) (and this is clearly not one to one). To show that A is "onto", you need to show that for all y in $$\mathbb{R}^2$$,there exist x in $$\mathbb{R}^3$$ so that Ax= y. The simplest way to do that is to row-reduce A. If the second row does not reduce to all 0s, it's true.