To find the image of the line segment $(u, v) = (0, 0)$ to $(u, v) = (2, 1)$ under the transformation $T$ defined by:

$x = u^2 - v^2 \quad \text{and} \quad y = 2uv,$

we need to determine how points along this line segment in $(u, v)$-space map to points in $(x, y)$-space.

Step 1: Parametrize the Line Segment in $(u, v)$-space

The line segment starts at $(u, v) = (0, 0)$ and ends at $(u, v) = (2, 1)$. We can parametrize it by setting: $u = 2t \quad \text{and} \quad v = t,$ where $t$ ranges from $0$ to $1$. This gives us the entire line segment in terms of $t$.

Step 2: Substitute into the Transformation Equations

Substitute $u = 2t$ and $v = t$ into the transformation equations for $x$ and $y$:

1. For $x$: $x = u^2 - v^2 = (2t)^2 - (t)^2 = 4t^2 - t^2 = 3t^2.$

2. For $y$: $y = 2uv = 2 \cdot (2t) \cdot t = 4t^2.$

So, the image of the line segment in $(x, y)$-space is given by the parametric equations: $x = 3t^2, \quad y = 4t^2, \quad \text{for } 0 \leq t \leq 1.$

Step 3: Eliminate the Parameter $t$

To express the curve in $(x, y)$-space without $t$, solve for $t^2$ in terms of $x$: $t^2 = \frac{x}{3}.$ Substitute this into the equation for $y$: $y = 4t^2 = 4 \left( \frac{x}{3} \right) = \frac{4}{3}x.$

Thus, the image of the line segment is the line $y = \frac{4}{3}x$ in $(x, y)$-space, with $x$ ranging from $0$ to $3$ (as $x = 3t^2$ when $t = 1$).

How to Sketch the Image

1. Draw the $x$- and $y$-axes.
2. Plot the starting point of the line segment in $(x, y)$-space: when $t = 0$, $(x, y) = (0, 0)$.
3. Plot the endpoint: when $t = 1$, $(x, y) = (3, 4)$.
4. Draw a line segment connecting $(0, 0)$ and $(3, 4)$ to represent the image of the original line segment under the transformation.

Would you like a detailed step-by-step solution on any part of this process or more visualization tips?

Further Questions to Explore

1. What happens to the image of the line segment when the endpoints change?
2. How does the transformation behave for other line segments in $(u, v)$-space?
3. Can we find the inverse of this transformation?
4. What does the Jacobian of this transformation tell us about area scaling?
5. How do the lines $u = v$ and $u = -v$ map under this transformation?

Tip

When working with transformations, parametrizing curves in terms of a single variable makes it easier to track changes under the transformation and analyze the resulting shape in the target space.