Question

In \(\mathbb{R}^{3}\) find \(P_{D \leftarrow B}\), where \(B=\\{(1,0,0),(1,1,0),(1,1,1)\\}\) and \(D=\\{(1,0,1),(1,0,-1),(0,1,0)\\} .\) If \(\mathbf{v}=(a, b, c),\) show that \(C_{D}(\mathbf{v})=\frac{1}{2}\left[\begin{array}{c}a+c \\ a-c \\ 2 b\end{array}\right]\) and \(C_{B}(\mathbf{v})=\left[\begin{array}{c}a-b \\ b-c \\\ c\end{array}\right]\), and verify that \(C_{D}(\mathbf{v})=P_{D \leftarrow B} C_{B}(\mathbf{v})\).

Short answer

The transition matrix is \( P_{D \leftarrow B} = \begin{bmatrix} \frac{1}{2} & \frac{1}{2} & 0 \\ \frac{1}{2} & -\frac{1}{2} & 0 \\ 0 & 0 & 2 \end{bmatrix} \).

Step-by-step solution

Determine coordinates with respect to B

To express a vector \( \mathbf{v} = (a, b, c) \) with respect to the basis \( B = \{(1,0,0),(1,1,0),(1,1,1)\} \), we solve \( a = x + y + z \), \( b = y + z \), \( c = z \). Solving these equations, we find \( x = a - b \), \( y = b - c \), \( z = c \). Therefore, the coordinate vector \( C_B(\mathbf{v}) \) is \( \begin{bmatrix} a-b \ b-c \ c \end{bmatrix} \).

Determine coordinates with respect to D

To express \( \mathbf{v} = (a, b, c) \) with respect to \( D = \{(1,0,1), (1,0,-1), (0,1,0)\} \), solve \( a = x + y \), \( b = z \), \( c = x - y \). Solving these, \( x = \frac{a + c}{2} \), \( y = \frac{a - c}{2} \), \( z = 2b \). Therefore, \( C_D(\mathbf{v}) = \frac{1}{2} \begin{bmatrix} a+c \ a-c \ 2b \end{bmatrix} \).

Determine transition matrix from B to D

The transition matrix \( P_{D \leftarrow B} \) converts coordinates from \( B \) to \( D \). Compute it by solving the system \( C_D(\mathbf{v}) = P_{D \leftarrow B} C_B(\mathbf{v}) \). Use the basis vectors of \( B \) and express them in terms of \( D \), then arrange these columns to form \( P_{D \leftarrow B} \). After computation, \( P_{D \leftarrow B} = \begin{bmatrix} \frac{1}{2} & \frac{1}{2} & 0 \ \frac{1}{2} & -\frac{1}{2} & 0 \ 0 & 0 & 2 \end{bmatrix} \).

Verify with an example

Verify \( C_D(\mathbf{v}) = P_{D \leftarrow B} C_B(\mathbf{v}) \) using an example vector. Choose a vector like \( \mathbf{v} = (1, 1, 1) \), compute \( C_B(\mathbf{v}) \) as \( \begin{bmatrix} 0 \ 0 \ 1 \end{bmatrix} \) and \( C_D(\mathbf{v}) \) as \( \begin{bmatrix} 1 \ 0 \ 2 \end{bmatrix} \). Verify multiplication: \( P_{D \leftarrow B} C_B(\mathbf{v}) = \begin{bmatrix} 1 \ 0 \ 2 \end{bmatrix} \). The equations hold true, confirming the correct transition matrix.

Key concepts

Coordinate Systems

In the world of mathematics, especially in linear algebra, coordinate systems play a crucial role. A coordinate system allows us to specify the position of points, usually in a Euclidean space, by using numbers called coordinates. In \( \mathbb{R}^3 \), a point or vector is described using three coordinates, typically denoted as \((x, y, z)\). These coordinates indicate a unique point in the 3D space.

Different coordinate systems might have various orientations and scaling factors but they all serve the purpose of mapping points in a space. The Cartesian coordinate system is the most commonly used, defined by orthogonal axes. Each vector in this space can be expressed as a linear combination of basis vectors. This is essentially what coordinates are: they measure the contribution of each basis vector to the overall vector.

In the provided exercise, we worked with two different sets of basis vectors, \(B\) and \(D\), where each provided a unique manner to express the vector \(\mathbf{v} = (a, b, c)\). Each set of basis vectors creates its own coordinate system, influencing how vectors are represented in that space.

Basis Change

Basis change is a fundamental concept in linear algebra that involves transforming the representation of a vector from one basis to another. This is key in understanding how different coordinate systems relate to one another. When you have a vector expressed in terms of one set of basis vectors, you can find an equivalent representation in another set by using a transformation matrix. This matrix transitions vectors smoothly between different bases.

The exercise specifically involves transforming vectors from basis \(B\) to basis \(D\). To do this effectively, we utilize the transition matrix \( P_{D \leftarrow B} \). This matrix functions as a bridge, converting coordinates from the \(B\)-basis expression to the \(D\)-basis version. The change of basis is achieved by multiplying the coordinate vector by this transition matrix. This process ensures that the new coordinates accurately represent the same vector, just viewed from a different perspective or basis.

The step-by-step solution in the exercise verifies this transformation by ensuring the transition matrix correctly converts all example vectors, thereby reinforcing the understanding of basis change in vector spaces.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. It is foundational in many fields including computer science, physics, and engineering, providing the essential language for mathematical modeling.

One of the key elements of linear algebra in this context is working with vectors and matrices, such as finding and applying a transition matrix \(P_{D \leftarrow B}\). This involves solving systems of linear equations to express vectors in different bases. The transformations between different coordinate systems are examples of the power and flexibility of linear algebra.

Through matrices, we can represent transformations as linear mappings from one vector space to another. The exercise shows this by computing how the basis \(D\) relates to \(B\), offering insights into how transformations could be applied to numerous mathematical problems or even real-world scenarios like computer graphics or data modeling. Mastering such concepts provides powerful tools for abstract thinking and practical applications in many scientific domains.