Range and Image of a Transformation

In summary, the problem is to find an $\textbf{x}$ whose image under the transformation is $\textbf{b}$.If $A$ is the same and $\mathbf{b}=\begin{bmatrix}\dfrac{151}{6}\\ -21\\ -\dfrac{37}{2}\\ -\dfrac{50}{3}\end{bmatrix}$, then $\mathbf{b}$ is in the range of $\textbf{x} \mapsto A\textbf{x}$. However, if $A$ is the same and $\mathbf{b}=\begin{bmatrix}\dfrac{151}{6}
  • #1
bwpbruce
60
1
$\textbf{Problem}$

Let $\textbf{b} = \begin{bmatrix}\begin{array}{r} 8 \\ 7 \\ 5 \\ -3 \end{array}\end{bmatrix}$ and let $A = \begin{bmatrix} 2 & 3 & 5 & - 5 \\ -7 & 7 & 0 & 0 \\ -3 & 4 & 1 & 3 \\ -9 & 3 & -6 & -4 \end{bmatrix}$

Is $\textbf{b}$ in the range of the transformation $\textbf{x} \mapsto A\textbf{x}$. If so, find an $\textbf{x}$ whose image under the transformation is $\textbf{b}$.$\textbf{My Solution}$
By inspection, notice that the sum of columns $\textbf{a}_2$ and $\textbf{a}_3$ is $\textbf{b}$.

In other words: \begin{align*}\begin{bmatrix} 3 \\ 7 \\ 4 \\ 3 \end{bmatrix} + \begin{bmatrix} 5 \\ 0 \\ 1 \\ -6 \end{bmatrix} &= \begin{bmatrix} 3 + 5 \\ 7 + 0 \\ 4 + 1 \\ 3 - 6 \end{bmatrix} = \begin{bmatrix} 8 \\ 7 \\ 5 \\ -3 \end{bmatrix} \end{align*}
So $\textbf{b}$ is in the range of $T$.
Also by inspection, it is obvious that for $A\textbf{(x)} = \textbf{b}$, $\textbf{x} = \begin{bmatrix} 0 \\ 1 \\ 1 \\ 0 \end{bmatrix}$.

Is my approach to this acceptable?
 
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  • #2
Is the element of $A$ in row 3, column 2 equal to $-4$ or $4$? If it is $-4$, then $\mathbf{a}_2+\mathbf{a}_3\ne\mathbf{b}$.

If that element is 4, then your solution is correct, but what would you do if you couldn't guess an answer?
 
  • #3
Evgeny.Makarov said:
Is the element of $A$ in row 3, column 2 equal to $-4$ or $4$? If it is $-4$, then $\mathbf{a}_2+\mathbf{a}_3\ne\mathbf{b}$.

If that element is 4, then your solution is correct, but what would you do if you couldn't guess an answer?

You're good at finding typos. What do you mean by if you $\textbf{couldn't}$ guess an answer? I think I know this "other way" you're referring to. But I'm just saying... Show me a situation where $\textbf{b}$ is in the range but one is forced to use the alternative method rather than the method I posted above.
 
  • #4
Let $A$ be the same and $\mathbf{b}=\begin{bmatrix}\dfrac{151}{6}\\ -21\\ -\dfrac{37}{2}\\ -\dfrac{50}{3}\end{bmatrix}$. Is $\mathbf{b}$ in the range of $\mathbf{x}\mapsto A\mathbf{x}$?
 
  • #5
That's clever. The alternative method is to create a matrix of the form $\begin{bmatrix}A & \textbf{b} \end{bmatrix}$ then simplify to reduced echelon form. If the system is consistent then $\textbf{b}$ is in the range of $\textbf{x} \mapsto A\textbf{x}$.
 
  • #6
Yes.
 
  • #7
Well, for the given problem, my method was more efficient.
 

FAQ: Range and Image of a Transformation

What is the difference between the range and image of a transformation?

The range of a transformation is the set of all possible outputs or values that the transformation can produce. The image of a transformation is the set of all actual outputs or values that are obtained by applying the transformation to a specific set of inputs.

How can you determine the range and image of a transformation?

To determine the range and image of a transformation, you can either algebraically manipulate the transformation equation or visually graph the transformation and observe the output values.

Can the range and image of a transformation be the same?

Yes, the range and image of a transformation can be the same if the transformation is one-to-one and onto, meaning that each input value corresponds to a unique output value and every possible output value is reached.

What is the significance of the range and image of a transformation?

The range and image of a transformation provide insight into the behavior and limitations of the transformation. They can also be used to determine if a transformation is invertible or to identify the nature of the transformation, such as linear or nonlinear.

Can the range and image of a transformation change?

Yes, the range and image of a transformation can change depending on the inputs or conditions of the transformation. For example, if the transformation involves a constant that can be varied, the range and image may also vary.

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