Finding Linear Transformation T R^2 To R^3 T(x, Y) Explained

Jul 14, 2025 by ADMIN 61 views

Linear Transformation T R^2 to R^3 Article

In linear algebra, understanding linear transformations is crucial, and in this article, we will delve deep into a specific problem involving a linear transformation T from R² to R³. This is a fundamental concept in mathematics, especially when dealing with vector spaces and their mappings. We are given two specific transformations: T(1, 2) = (1, 3, 2) and T(1, 0) = (1, -1, 0), and our objective is to find a general expression for T((x, y)). This problem not only solidifies our understanding of linear transformations but also showcases how to leverage the properties of linearity to solve such problems. We will explore the underlying principles, the step-by-step methodology, and the theoretical justifications to ensure a comprehensive grasp of the topic. By the end of this article, you should be able to tackle similar problems with confidence, applying the techniques and insights discussed herein. This exploration will also touch upon the significance of linear transformations in various fields such as computer graphics, physics, and engineering, where they play a pivotal role in modeling and solving complex problems.

Before we dive into solving the specific problem, it is crucial to understand what linear transformations are and the properties that define them. A linear transformation, often denoted as T, is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. Mathematically, this means that for any vectors u and v in the domain vector space, and any scalar c, the following two conditions must hold:

T(u + v) = T(u) + T(v) (Additivity)
T(cu) = cT(u) (Homogeneity)

These two conditions encapsulate the essence of linearity. The additivity property ensures that the transformation of the sum of two vectors is the sum of their individual transformations. The homogeneity property ensures that scaling a vector before transformation is the same as scaling the transformed vector. These properties are not just abstract mathematical rules; they have profound implications in how linear transformations behave and how we can manipulate them to solve problems.

In the context of our problem, T: R² → R³, this means that T takes vectors from the two-dimensional Euclidean space (R²) and maps them to vectors in the three-dimensional Euclidean space (R³). The fact that T is linear allows us to break down the transformation of any vector in R² into a linear combination of the transformations of basis vectors. This is a powerful technique that we will use to find the general expression for T((x, y)).

The given transformations, T(1, 2) = (1, 3, 2) and T(1, 0) = (1, -1, 0), provide us with specific instances of how T acts on two particular vectors. The challenge is to use this information, along with the properties of linearity, to deduce the transformation rule for any arbitrary vector (x, y) in R². This involves expressing (x, y) as a linear combination of (1, 2) and (1, 0), or other suitable vectors, and then applying the linearity properties to find T((x, y)). Understanding these foundational concepts is key to mastering linear algebra and its applications.

To tackle the problem of finding T((x, y)), we must effectively use the given information: T(1, 2) = (1, 3, 2) and T(1, 0) = (1, -1, 0). The core idea is to express an arbitrary vector (x, y) in R² as a linear combination of the vectors (1, 2) and (1, 0). This is a crucial step because if we can write (x, y) as a(1, 2) + b(1, 0) for some scalars a and b, then, due to the linearity of T, we have:

T((x, y)) = T(a(1, 2) + b(1, 0))

Using the properties of linear transformations, we can further expand this as:

T((x, y)) = aT(1, 2) + bT(1, 0)

Since we know T(1, 2) and T(1, 0), the problem reduces to finding the scalars a and b in terms of x and y. This is a standard technique in linear algebra, where we leverage the fact that linear transformations preserve linear combinations. Expressing a vector as a linear combination of other vectors allows us to use the known transformations of those vectors to find the transformation of the original vector.

The equation (x, y) = a(1, 2) + b(1, 0) can be rewritten as a system of linear equations:

x = a + b
y = 2a

Solving this system for a and b will give us the coefficients needed to express (x, y) as a linear combination of (1, 2) and (1, 0). Once we have a and b, we can substitute them back into the equation T((x, y)) = aT(1, 2) + bT(1, 0) to find the general expression for T((x, y)). This approach highlights the power of using linearity to break down complex transformations into simpler, manageable components. The ability to manipulate vectors and transformations in this way is a cornerstone of linear algebra and its applications.

To express (x, y) as a linear combination of (1, 2) and (1, 0), we need to solve the system of linear equations we derived in the previous section. The system is:

x = a + b
y = 2a

This is a straightforward system that can be solved using various methods, such as substitution or elimination. Here, we will use the substitution method for clarity and efficiency. From the second equation, y = 2a, we can directly solve for a:

a = y/2

Now that we have a in terms of y, we can substitute this into the first equation, x = a + b, to solve for b:

x = (y/2) + b
b = x - (y/2)

Thus, we have found expressions for both a and b in terms of x and y:

a = y/2
b = x - y/2

These coefficients are crucial because they allow us to write the vector (x, y) as a specific linear combination of the vectors (1, 2) and (1, 0). Specifically, we have:

(x, y) = (y/2)(1, 2) + (x - y/2)(1, 0)

This expression is the key to finding T((x, y)). By expressing (x, y) in this form, we can now apply the linearity of T and the given transformations T(1, 2) and T(1, 0) to find the general transformation rule. This process demonstrates the power of breaking down a problem into smaller, manageable parts and using the properties of linear transformations to our advantage. The ability to solve for these coefficients efficiently is a fundamental skill in linear algebra, with applications in various areas such as computer graphics, optimization, and data analysis.

Now that we have expressed (x, y) as a linear combination of (1, 2) and (1, 0), we can apply the linear transformation T to find T((x, y)). Recall that we have:

(x, y) = (y/2)(1, 2) + (x - y/2)(1, 0)

Using the linearity of T, we can write:

T((x, y)) = T[(y/2)(1, 2) + (x - y/2)(1, 0)]

Applying the additivity and homogeneity properties of linear transformations, we get:

T((x, y)) = (y/2)T(1, 2) + (x - y/2)T(1, 0)

We are given that T(1, 2) = (1, 3, 2) and T(1, 0) = (1, -1, 0). Substituting these values into the equation, we have:

T((x, y)) = (y/2)(1, 3, 2) + (x - y/2)(1, -1, 0)

Now, we perform the scalar multiplication and vector addition:

T((x, y)) = (y/2, 3y/2, y) + (x - y/2, -x + y/2, 0)

Finally, we add the vectors component-wise:

T((x, y)) = (y/2 + x - y/2, 3y/2 - x + y/2, y + 0)

Simplifying, we obtain the final expression for T((x, y)):

T((x, y)) = (x, 2y - x, y*)

This is the general transformation rule for the given linear transformation T. It shows how any vector (x, y) in R² is mapped to a vector in R³ under the transformation T. This result highlights the elegance and power of linear algebra, where we can use a few known transformations to deduce the transformation rule for any vector in the space. The ability to derive such general rules is crucial in various applications, including computer graphics, physics, and engineering, where linear transformations are used to model and solve complex problems.

After carefully applying the principles of linear transformations and solving the associated equations, we have arrived at the general expression for T((x, y)). The final result is:

T((x, y)) = (x, 2y - x, y)

This equation tells us exactly how the linear transformation T maps any vector (x, y) from R² to R³. The transformation takes the x-component of the input vector and leaves it unchanged as the first component of the output vector. The second component of the output vector is a linear combination of the x and y components of the input vector, specifically 2y - x. The third component of the output vector is simply the y-component of the input vector.

This result encapsulates the essence of the problem and demonstrates the power of linear algebra in solving such transformations. By leveraging the properties of linearity, we were able to break down the problem into manageable steps: expressing (x, y) as a linear combination of basis vectors, applying the transformation to this linear combination, and then simplifying the resulting expression. This approach is not only effective but also provides a clear and systematic way to tackle similar problems in the future.

The transformation T has some interesting geometric interpretations. It shears the plane in the x-y direction and then embeds it into R³. The x-coordinate remains the same, while the y-coordinate is transformed in a way that depends on both x and y. The z-coordinate is simply the original y-coordinate. This kind of geometric understanding is invaluable in many applications, such as computer graphics and robotics, where linear transformations are used to manipulate objects in space.

In conclusion, the ability to find general transformation rules like this is a fundamental skill in linear algebra. It allows us to understand and manipulate vector spaces and their mappings, which are essential tools in various scientific and engineering disciplines. The process we have followed here—breaking down the problem, using linearity, and solving for coefficients—is a general strategy that can be applied to a wide range of linear transformation problems.

In summary, we have successfully determined the linear transformation T(x, y) given the transformations T(1, 2) = (1, 3, 2) and T(1, 0) = (1, -1, 0). The key steps involved expressing an arbitrary vector (x, y) as a linear combination of the vectors (1, 2) and (1, 0), solving for the coefficients of this linear combination, and then applying the linearity properties of T to find the general expression:

T((x, y)) = (x, 2y - x, y)

This process demonstrates the core principles of linear algebra, particularly the use of linear combinations and the properties of linear transformations. The ability to manipulate vectors and transformations in this manner is fundamental to many areas of mathematics, science, and engineering. Linear transformations are not just abstract mathematical concepts; they are powerful tools that allow us to model and solve a wide range of problems in various fields.

For instance, in computer graphics, linear transformations are used to rotate, scale, and translate objects in 3D space. In physics, they are used to describe changes of coordinates and transformations of physical quantities. In machine learning, they are used in dimensionality reduction techniques and in the construction of neural networks. The applications are vast and varied, highlighting the importance of understanding linear transformations.

Furthermore, the problem-solving approach we have used in this article is applicable to many other problems in linear algebra. The general strategy of expressing vectors as linear combinations of basis vectors and using the properties of linearity is a powerful technique that can be used to solve a wide range of problems. This includes finding the matrix representation of a linear transformation, determining the kernel and image of a linear transformation, and solving systems of linear equations.

In conclusion, mastering linear transformations is essential for anyone studying mathematics, science, or engineering. The concepts and techniques we have explored in this article provide a solid foundation for further study and application in these fields. The ability to think linearly, to manipulate vectors and transformations, and to solve linear problems is a valuable skill that will serve you well in your academic and professional endeavors.