Question

Given \(u=\left[\begin{array}{r}10 \\ 6\end{array}\right]\) and \(v=\left[\begin{array}{l}4 \\ 8\end{array}\right],\) which of the following are convex combinations of \(u\) \((a)\left[\begin{array}{l}7 \\ 7\end{array}\right]\) \((b)\left[\begin{array}{l}5.2 \\ 7.6\end{array}\right]\) \((c)\left[\begin{array}{l}6.2 \\ 8.2\end{array}\right]\)

Short answer

All options (a), (b), and (c) are convex combinations of vectors \(u\) and \(v\).

Step-by-step solution

Understanding a Convex Combination

A convex combination of two vectors \(u\) and \(v\) can be expressed in the form \(w = alpha u + \beta v\) where \(\alpha + \beta = 1\) and both \(\alpha\) and \(\beta\) are non-negative coefficients. We will determine if each vector \(w\) satisfies these conditions for given \(u\) and \(v\).

Calculate for Option (a)

We need to check if \(\begin{bmatrix} 7 \ 7 \end{bmatrix}\) is a convex combination of \(u\) and \(v\). Let \(alpha u + \beta v = \begin{bmatrix} 7 \ 7 \end{bmatrix}\). This translates to \( \begin{bmatrix} 10alpha + 4\beta \ 6alpha + 8\beta \end{bmatrix} = \begin{bmatrix} 7 \ 7 \end{bmatrix}\).Solving these equations:1. \(10alpha + 4\beta = 7\)2. \(6alpha + 8\beta = 7\)3. \(\alpha + \beta = 1\)Check if \(\alpha, \beta \) are non-negative and satisfy properties.

Solution Calculation

Solving for \(\alpha + \beta = 1\), substitute \(\beta = 1 - \alpha\):1. From first equation: \[ 10alpha + 4(1 - alpha) = 7 \to 10alpha + 4 - 4alpha = 7 \to 6alpha = 3 \to alpha = 0.5 \]2. Substitute \(\alpha = 0.5\) into equation 3: \[ \beta = 1 - 0.5 = 0.5 \]both are non-negative, verify: - 6n\alpha + 8\beta = 6(0.5) + 8(0.5) = 3 + 4 = 7 which is consistent. Yes, (a) is a convex combination.

Check Option (b)

Now check if \(\begin{bmatrix} 5.2 \ 7.6 \end{bmatrix}\) is a convex combination of \(u\) and \(v\). Let \(alpha u + \beta v = \begin{bmatrix} 5.2 \ 7.6 \end{bmatrix}\). The equations are:1. \(10alpha + 4\beta = 5.2\)2. \(6alpha + 8\beta = 7.6\)3. \(\alpha + \beta = 1\)Solve and check conditions.

Solution Calculation

Substitute \(\beta = 1 - \alpha\) into the first equation:1. \[ 10alpha + 4(1 - alpha) = 5.2 \to 10alpha + 4 - 4alpha = 5.2 \to 6alpha = 1.2 \to alpha = 0.2 \]2. Substitute into equation 3: \[ \beta = 1 - 0.2 = 0.8 \]both non-negative.Verify using the second equation: - 6n\alpha + 8\beta = 6(0.2) + 8(0.8) = 1.2 + 6.4 = 7.6. Consistent. Yes (b) is a convex combination.

Test Option (c)

Finally, check \(\begin{bmatrix} 6.2 \ 8.2 \end{bmatrix}\) for convex combination. Let \(alpha u + \beta v = \begin{bmatrix} 6.2 \ 8.2 \end{bmatrix}\) giving:1. \(10alpha + 4\beta = 6.2\)2. \(6alpha + 8\beta = 8.2\)3. \(\alpha + \beta = 1\)Solve and determine validity.

Solution Calculation

Substitute \(\beta = 1 - \alpha\):1. From first equation: \[ 10alpha + 4(1 - alpha) = 6.2 \to 10alpha + 4 - 4alpha = 6.2 \to 6alpha = 2.2 \to alpha = \frac{2.2}{6} \approx 0.3667 \]2. Thus \(\beta = 1 - 0.3667 \approx 0.6333\) both non-negative. Verify:- \(6alpha + 8\beta = 6(0.3667) + 8(0.6333) \approx 2.2 + 5.0664 \approx 8.2\).This is valid. Yes (c) is also a convex combination.

Key concepts

Vectors

Vectors are essential tools in mathematics, particularly in fields such as physics and engineering. They are entities with both magnitude and direction, representing various physical quantities.

• A vector is denoted by an arrow, where the length represents the magnitude and the point indicates direction.
• In a coordinate system, vectors are often represented by coordinates, such as \( u = \begin{bmatrix} 10 \ 6 \end{bmatrix} \) and \( v = \begin{bmatrix} 4 \ 8 \end{bmatrix} \).
• Vectors can be added together by adding their corresponding components, which can result in a new vector.
• Scalar multiplication involves multiplying a vector by a real number, scaling its magnitude while keeping its direction.

Understanding vectors is crucial for tackling problems involving directions and magnitudes, like those in physics or computer graphics.

Linear Algebra

Linear Algebra is a branch of mathematics dealing with vector spaces and linear equations. It plays a pivotal role in various areas like computer science, engineering, and natural sciences.

• At its core, linear algebra is concerned with linear transformations and systems of linear equations.
• The study involves matrices, which are arrays of numbers representing vectors in systems of equations.
• Operations such as addition, subtraction, and multiplication of matrices lay the foundation for solving complex systems.
• The concept of a convex combination originates from linear algebra, referring to linear parameters \( \alpha \) and \( \beta \) satisfying \( \alpha + \beta = 1\).

Mastering linear algebra opens the door to applications in machine learning, data science, and optimization.

Matrix Equations

Matrix equations are utilized to express and solve systems of linear equations in a compact form. They simplify complex calculations that involve multiple vectors and their transformations.

• A matrix is a two-dimensional array of numbers often used to represent a system of linear equations.
• By organizing coefficients in matrices, solving these systems becomes more systematic and efficient.
• To determine if a particular solution is a convex combination, we use matrices to express equations in terms of vectors.
• Matrix equations allow for scalable solutions, making them suitable for computational tasks and algorithm implementations.

In linear algebra, matrix equations are fundamental for analyzing vector transformations and optimizing linear systems.