Q. 53 - Calculus

Question

Let α, β, γ , and δ be constants. A transformation$T: R^2\Rightarrow R^2$ where $x=\alpha u+\beta v$ and $y=\gamma u+\delta v$, is called a linear transformation of $R^2$. Use this transformation to answer Exercises 53–55.

Prove that a linear transformation takes a line ax + by = c in the XY-plane to a line in the UV-plane if the Jacobian of the transformation is nonzero. 

Step-by-Step Solution

Answer

It is proven that a linear transformation takes a line ax + by = c in the XY-plane to a line in the UV-plane if the Jacobian of the transformation is nonzero. 

Step 1: Given information

The equations of transformations are,

$x=\alpha u+\beta v; y=\gamma u+\delta v$

The objective is to determine the transformation of a line $ax+by=c$ in xy-plane to uv-plane.

Step 2: Proof

The definition of a Jacobian of transformation using partial derivatives is given as

$$\begin{gathered} \frac{\partial (x, y)}{\partial (u, v)}=\left|\begin{array}{cc}\frac{\partial x}{\partial u}& \frac{\partial y}{\partial u}\\ \frac{\partial x}{\partial v}& \frac{\partial x}{\partial v}\end{array}\right| \\ \frac{\partial (x, y)}{\partial (u, v)}=\left|\begin{array}{cc}\alpha & \gamma \\ \beta & \delta \end{array}\right| \\ \frac{\partial (x, y)}{\partial (u, v)}=\alpha \delta -\beta \gamma \end{gathered}$$

To form the equation of the line in UV- plane , substitute the equations of transformations in the equation of line.

$$\begin{gathered} ax+by=c \\ (a\alpha +b\gamma )u+(a\beta +b\delta )v=c \end{gathered}$$

If the above equation has to represent a line in uv- planes, both the coefficients of variables u and v have to be non-zero.

Let's assume the converse.

$$\begin{gathered} a\alpha +b\gamma =0 \\ a\beta +b\delta =0 \end{gathered}$$

Consider the ratio of two statements.

$$\begin{gathered} \frac{\alpha }{\beta }=\frac{\gamma }{\delta } \\ \alpha \delta -\beta \gamma =0 \end{gathered}$$

Thus, it is proved that if the Jacobian of transformation is equal to zero, then the coefficients of both variables u and v in the transformed equation are also zero.

Hence, A linear transformation takes a line ax + by = c in the XY-plane to a line in the UV plane if the Jacobian of the transformation is nonzero.