Transition Matrix From Basis B To B' And Vector Representation

In linear algebra, understanding how to change bases is crucial for solving various problems. This involves finding the transition matrix, which allows us to convert the coordinates of a vector from one basis to another. This article delves into the process of finding the transition matrix from basis B to basis B' and then uses this matrix to determine the vector x represented in the standard basis. We will explore the significance of basis transformations and how they simplify calculations and provide insights into the underlying vector space structure. Our main focus will be on the mechanics of constructing the transition matrix and its application in converting coordinate vectors. Understanding these concepts is fundamental for anyone working with vector spaces and linear transformations. We will break down each step in detail, ensuring clarity and comprehension. Additionally, we will highlight the importance of these techniques in various applications such as computer graphics, data analysis, and engineering.

Given two bases, B = {(3, 2), (1, 1)} and B' = {(-12, 0), (-4, 4)}, and the coordinate vector of x with respect to B', which is [x]_B' = [-1, 3]^T, we aim to:

1. Find the transition matrix from B to B'. This matrix, denoted as P^-1, will allow us to convert coordinate vectors from the B basis to the B' basis.
2. Determine the vector x in the standard basis using the given information.

This problem showcases a practical application of basis transformation. The transition matrix acts as a bridge between different representations of the same vector, providing a powerful tool for simplifying calculations and gaining deeper insights. This process is frequently used in various fields, including computer graphics, engineering, and data science, where changing bases can reveal hidden structures or simplify complex computations. By understanding how to construct and use transition matrices, we can efficiently navigate different coordinate systems and manipulate vectors in a more intuitive way.

Part (a): Finding the Transition Matrix from B to B'

The transition matrix from B to B', denoted as P^-1, transforms the coordinates of a vector in the B basis to its coordinates in the B' basis. To find P^-1, we express the vectors in B as linear combinations of the vectors in B'. Let B = {v₁, v₂} = {(3, 2), (1, 1)} and B' = {u₁, u₂} = {(-12, 0), (-4, 4)}. We need to find scalars a, b, c, and d such that:

v₁ = au₁ + bu₂

(3, 2) = a(-12, 0) + b(-4, 4)

This gives us the system of equations:

3 = -12a - 4b

2 = 4b

Solving for b from the second equation:

b = 2 / 4 = 0.5

Substituting b into the first equation:

3 = -12a - 4(0.5)

3 = -12a - 2

5 = -12a

a = -5 / 12

So, v₁ = (-5/12)u₁ + (1/2)u₂. Now, we express v₂ as a linear combination of u₁ and u₂:

v₂ = cu₁ + du₂

(1, 1) = c(-12, 0) + d(-4, 4)

This gives us the system of equations:

1 = -12c - 4d

1 = 4d

Solving for d from the second equation:

d = 1 / 4 = 0.25

Substituting d into the first equation:

1 = -12c - 4(0.25)

1 = -12c - 1

2 = -12c

c = -2 / 12 = -1 / 6

So, v₂ = (-1/6)u₁ + (1/4)u₂. The transition matrix P^-1 is formed by using the coefficients a, b, c, and d as columns:

P^-1 =

| -5/12 -1/6 |

| 1/2 1/4 |

This transition matrix P^-1 allows us to convert the coordinates of a vector from the B basis to the B' basis. Each column of P^-1 represents the coordinates of the corresponding vector from basis B expressed in terms of basis B'. This matrix is fundamental for changing representations and understanding how vectors transform under different coordinate systems.

Part (b): Finding the Vector x in the Standard Basis

To find the vector x in the standard basis, given its coordinate vector in B' as [x]_B' = [-1, 3]^T, we first express x as a linear combination of the basis vectors in B'. Let B' = {u₁, u₂} = {(-12, 0), (-4, 4)}. Then,

x = -1u₁ + 3u₂

x = -1**(-12, 0)** + 3**(-4, 4)**

x = (12, 0) + (-12, 12)

x = (0, 12)

Therefore, the vector x in the standard basis is (0, 12). This process illustrates how coordinate vectors can be transformed back into their original vector representation using the basis vectors. Understanding this transformation is essential for working with vectors in different coordinate systems and for applying linear transformations.

In this article, we successfully found the transition matrix P^-1 from basis B to basis B', which is:

P^-1 =

| -5/12 -1/6 |

| 1/2 1/4 |

We then used the coordinate vector of x in the B' basis, [x]_B' = [-1, 3]^T, to determine the vector x in the standard basis, which is (0, 12). These calculations demonstrate the importance of transition matrices in changing between different vector space representations. The transition matrix acts as a bridge, allowing us to easily convert coordinate vectors from one basis to another. This is a fundamental concept in linear algebra with broad applications across various fields. Understanding basis transformations is crucial for simplifying calculations, gaining insights into the structure of vector spaces, and applying linear algebra techniques in practical scenarios. This process is not only vital for theoretical understanding but also for practical applications in fields such as computer graphics, data analysis, and engineering, where manipulating vectors and coordinate systems is a common task.