## The range of a matrix transformation

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### The range of a matrix transformation

How to do you determine if a vector (w) is in the range a matrix transformation.

Let f: R^2 --> R^3
f(x) = Ax

$\ A=\begin{bmatrix} 1 &2 \\ 0 &1 \\ 1& 1 \end{bmatrix}$/extract_itex] Vector w $\ w=\begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix}\$ Why is it a no? My work: ( I don't know if I'm doing it right) $\ f(x)=\begin{bmatrix} 1 &2 \\ 0 &1 \\ 1 & 1 \end{bmatrix}\begin{bmatrix} x\\ y \end{bmatrix}\$ $\ =\begin{bmatrix} 1 &2 \\ 0& 1 \end{bmatrix}\begin{bmatrix} 1\\ 1 \end{bmatrix}=\begin{bmatrix} 3\\ 1 \end{bmatrix}\$ and why is for Vector w, it's a yes? $\ w=\begin{bmatrix} 8\\ 5\\ 3 \end{bmatrix}\$ $\ =\begin{bmatrix} 1 &2 \\ 0& 1 \end{bmatrix}\begin{bmatrix} 8\\ 5 \end{bmatrix}=\begin{bmatrix} 18\\ 5 \end{bmatrix}\$ The book says that the set of all images of the vectors in R^n is called the range of f. I don't know what it means. Can someone please explain to me how to determine the range. Thanks in advance remeday86 Queen of Hearts Posts: 76 Joined: Mon Feb 16, 2009 5:15 pm ### Re: The range of a matrix transformation You want w= f(v) = Av, where v is a vector $\[\begin{array}x\\ y\end{array}$$.

The equation should be:

$$\begin{array}1\\ 1\\ 1\end{array}$ = $\begin{array} 1&2\\ 0&1\\ 1&1\end{array}$ $\begin{array}x\\ y\end{array}$$

You cannot substitute values of w into v, you need to see if any value of v, when multiplied by A gives you w. See if there is any v=[x,y] which satisfies this equation.

Edit: Some clarification. "The set of all images" is the set $\{f(v)|v\in \mathbb{R}^2\}$. So, if w is in the image, w=f(v) for some vector v in R^2. To check if such a vector exists, see if there is any vector which satisfies f(v)=w.
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