Problem 60 - Calculus Early Transcendentals

Question

Linear transformations Consider the linear transformation \(T\) in \(\mathbb{R}^{2}\) given by \(x=a u+b v, y=c u+d v,\) where \(a, b, c,\) and \(d\) are real numbers, with \(a d \neq b c\) a. Find the Jacobian of \(T\) b. Let \(S\) be the square in the \(u v\) -plane with vertices (0,0) \((1,0),(0,1),\) and \((1,1),\) and let \(R=T(S) .\) Show that \(\operatorname{area}(R)=|J(u, v)|\) c. Let \(\ell\) be the line segment joining the points \(P\) and \(Q\) in the uv- plane. Show that \(T(\ell)\) (the image of \(\ell\) under \(T\) ) is the line segment joining \(T(P)\) and \(T(Q)\) in the \(x y\) -plane. (Hint: Use vectors.) d. Show that if \(S\) is a parallelogram in the \(u v\) -plane and \(R=T(S),\) then \(\operatorname{area}(R)=|J(u, v)| \operatorname{area}(S) .\) (Hint: Without loss of generality, assume the vertices of \(S\) are \((0,0),(A, 0)\) \((B, C),\) and \((A+B, C),\) where \(A, B,\) and \(C\) are positive, and use vectors.)

Step-by-Step Solution

Answer

In this exercise, we were given a linear transformation T in the form (x, y) = (au + bv, cu + dv). We have analyzed various properties of this transformation, such as its Jacobian, area of a transformed square, the transformed line segment, and the transformed parallelogram. We found that the area of a transformed parallelogram is the product of the absolute value of the Jacobian determinant and the area of the original parallelogram. These results can help us better understand the effects and applications of linear transformations in various fields, such as geometry, computer graphics, and engineering.

Step 1: Recall the definition of the Jacobian matrix

The Jacobian is a matrix representing the first-order partial derivatives of a given vector-valued function. In our case, the linear transformation T is given by (x, y) = (au + bv, cu + dv). We need to compute the Jacobian matrix using partial derivatives.

Step 2: Compute the partial derivatives

Compute the partial derivatives of x and y with respect to u and v: \(\frac{\partial x}{\partial u} = a,\frac{\partial x}{\partial v} = b,\frac{\partial y}{\partial u} = c,\frac{\partial y}{\partial v} = d\)

Step 3: Construct the Jacobian matrix

Arrange the partial derivatives in the Jacobian matrix format: \(J(u, v) = \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v}\\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\) b. Show that \(\operatorname{area}(R)=|J(u,v)|\):

Step 1: Calculate the area of S

The area of square S is 1 unit, since it has side lengths of 1.

Step 2: Compute the determinant of the Jacobian matrix

The determinant of J(u, v) represents the "area scaling factor" of the transformation. Compute it: \(|J(u, v)| = ad - bc\)

Step 3: Apply the transformation T to S

Applying T to the vertices of S, we obtain the vertices of R (the transformed square): \(T(0,0) = (0,0)\) \(T(1,0) = (a,c)\) \(T(0,1) = (b,d)\) \(T(1,1) = (a+b,c+d)\)

Step 4: Show that the area of R equals |J(u,v)|

Using the formula for the area of a parallelogram determined by its column vectors, the area of R is given by the absolute value of the determinant of the Jacobian matrix J(u,v). As computed earlier, this value is \(|ad - bc|\), which proves that \(\operatorname{area}(R) = |J(u,v)|\). c. Show that T(ℓ) is the line segment joining T(P) and T(Q):

Step 1: Use vectors to represent the transformation

Let P = (u1,v1) and Q = (u2,v2) be the two points connected by the line segment ℓ. Then, ℓ can be represented as: \(ℓ = (1-t)P + tQ\) Applying the transformation T to ℓ, we get: \(T(ℓ) = T((1-t)P + tQ)\)

Step 2: Show that T(ℓ) connects T(P) and T(Q)

Since T is a linear transformation, we can distribute T over the sum and scalar multiplication: \(T(ℓ) = (1-t)T(P) + tT(Q)\) This equation shows that T(ℓ) is the line segment joining T(P) and T(Q), as required. d. Show that if \(S\) is a parallelogram in the \(u v\) -plane and \(R=T(S),\) then \(\operatorname{area}\)(R)=|J(u, v)| \(\operatorname{area}(S)\):

Step 1: Note the vertices of S

Without loss of generality, assume that the vertices of S are: \((0,0), (A, 0), (B, C),\) and \((A+B, C)\)

Step 2: Find the area of S

Using the formula for the area of a parallelogram determined by its column vectors, the area of S is given by: \(\operatorname{area}(S) = |AC|\)

Step 3: Apply the transformation T to S

Applying T to the vertices of S, we obtain the vertices of R (the transformed parallelogram): \(T(0,0) = (0,0)\) \(T(A,0) = (aA,cA)\) \(T(B,C) = (aB+bC,cB+dC)\) \(T(A+B,C) = (a(A+B)+bC,c(A+B)+dC)\)

Step 4: Compute the area of R using vectors

Using the formula for the area of a parallelogram determined by its column vectors, the area of R is: \(\operatorname{area}(R) = |(aA)(cB+dC) - (cA)(aB+bC)|\) This simplifies to: \(\operatorname{area}(R) = |AC(ad - bc)|\) But ad - bc is the determinant of J, so we have: \(\operatorname{area}(R) = |J(u,v)|\operatorname{area}(S)\), which completes the proof.

Key Concepts

Jacobian Matrix

In the realm of calculus and linear algebra, the Jacobian matrix is a vital concept particularly when dealing with transformations. It's a rectangular array of numbers that express the gradient of a vector-valued function. In simpler terms, it holds all first-order partial derivatives of a function. For example, the linear transformation \( T \) in \( \mathbb{R}^2 \) given by the equations \( x = au + bv \) and \( y = cu + dv \) can be represented by a Jacobian matrix \( J(u, v) \). This matrix is constructed by arranging the partial derivatives as follows:
\[J(u, v) = \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix} = \begin{bmatrix} a & b \c & d \end{bmatrix}\]
Thus, understanding the Jacobian is crucial as it helps in identifying how functions change and transform variables across different dimensions.

Determinant

The determinant is a scalar value that is a property of a square matrix, such as the Jacobian matrix. It conveys crucial information about the matrix, including whether the matrix is invertible and the scaling factor it applies to areas or volumes. When considering a transformation such as \( T \), the determinant of its Jacobian matrix \( J(u, v) = \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is computed as:
\[|J(u, v)| = ad - bc\]
This value is particularly important because:

• It indicates the "stretch" or "compression" factor that the transformation \( T \) applies to units in space.
• A nonzero determinant indicates that the transformation is invertible, implying a meaningful transformation of areas from one plane to another.

Understanding the determinant helps in assessing the behaviors of linear transformations and how they affect shapes and sizes.

Parallelogram

A parallelogram is a four-sided figure with opposite sides parallel. In the context of linear transformations, it's common to use parallelograms to understand the concept of area distortion. If you imagine a square \( S \) in the \( uv \)-plane, when it undergoes transformation \( T \), its shape may alter to a parallelogram \( R \).
The area of \( R \) after this transformation can be related back to the Jacobian matrix, and is given by:
\[\operatorname{area}(R) = |J(u, v)| \operatorname{area}(S)\]
For example, the vertices of \( S \) can be mapped to new coordinates, forming the vertices of \( R \), thus creating a dynamic link between the geometric structures of squares and parallelograms. This concept is essential in studying distortions and maintaining proportions in space through linear transformations.

Linear Algebra

Linear Algebra is the branch of mathematics concerning vectors, matrices, and linear transformations. It forms the backbone of many practical applications including transformations and space manipulations. By using equations and transformations like \( T \), we delve into how dimensions interact.
Key components include:

• Vectors: Fundamental objects that can represent points in space, as well as directions and magnitudes.
• Matrices: Structures that organize numbers in rows and columns, serving crucial roles in transformations.
• Transformations: Functions that map one set of vectors to another, preserving linear structures.

In solving exercises involving linear transformations, linear algebra provides the tools and techniques necessary to interpret and solve complex mathematical challenges. Understanding how transformations affect shapes, areas, and spaces is central to numerous scientific and engineering fields.