Question

Given two vectors \(u\) and \(v\) in the 2 -space, find and sketch: (a) The set of all linear combinations of \(u\) and \(v\) (b) The set of all nonnegative linear combinations of \(u\) and \(v\) (c) The set of all convex combinations of \(u\) and \(v\)

Short answer

(a) The plane spanned by \(u\) and \(v\); (b) a cone starting at the origin; (c) the line segment between \(u\) and \(v\).

Step-by-step solution

Understanding Linear Combinations

A linear combination of two vectors \(u = (u_1, u_2)\) and \(v = (v_1, v_2)\) in 2-space involves two scalars, say \(a\) and \(b\). Any linear combination is given by the vector expression \(w = a\cdot u + b\cdot v = (a u_1 + b v_1, a u_2 + b v_2)\). Thus, the set of all linear combinations is the plane spanned by \(u\) and \(v\). In two-dimensional space, if \(u\) and \(v\) are not collinear, the set forms the whole 2D plane. If they are collinear, it forms a line.

Nonnegative Linear Combinations

Nonnegative linear combinations restrict the scalars \(a\) and \(b\) such that \(a \, \geq 0\) and \(b \, \geq 0\). The expression \(w = a\cdot u + b\cdot v\) now includes only those combinations where both coefficients are nonnegative. Graphically, this forms a conical region that originates at the origin and extends indefinitely into the first quadrant if \(u\) and \(v\) are neither zero nor collinear and located within the positive x- and y-axes.

Convex Combinations

A convex combination of vectors \(u\) and \(v\) requires that the scalars satisfy \(a \, \geq 0\), \(b \, \geq 0\), and \(a + b = 1\). Hence, the expression \(w = a\cdot u + b\cdot v\) results in a new vector that lies on the line segment joining \(u\) and \(v\). This is effectively the set of all such vectors that blend \(u\) and \(v\) proportionally while keeping their total contribution equal to 1.

Key concepts

Vectors

Vectors are mathematical objects used to represent quantities that have both magnitude and direction. In the context of 2-space, like in our problem, a vector such as \(u = (u_1, u_2)\) or \(v = (v_1, v_2)\) has two components: one in the horizontal direction (x-axis) and one in the vertical direction (y-axis).

Vectors can be visually represented as arrows in a two-dimensional grid, where the length of the arrow corresponds to the vector's magnitude, and the direction of the arrow denotes the vector's direction. They are essential in understanding linear combinations, as they are the building blocks for these combinations.

When discussing linear combinations, vectors are combined using scalar multiplication and vector addition, which allows us to explore spaces generated by the vectors through various combinations of their magnitudes and directions.

Convex Combinations

Convex combinations involve combining vectors in a way that the resulting vector lies within the boundaries formed by the original vectors. In mathematical terms, if you have vectors \(u\) and \(v\), any convex combination \(w\) can be represented as \(w = a \cdot u + b \cdot v\), where \(a \geq 0\), \(b \geq 0\), and \(a + b = 1\).

The constraint \(a + b = 1\) ensures that the resulting vector \(w\) is situated along the line segment joining \(u\) and \(v\). This is a crucial concept in geometry and optimization applications, as it highlights how combinations can be limited to means or midpoints that do not exceed the spatial area bounded by the vectors.

Convex combinations are fundamental in various fields such as computer graphics, economics, and physics, providing intuitive and practical solutions to problems involving linear interpolation and systems equilibrium.

Nonnegative Linear Combinations

Nonnegative linear combinations of vectors involve a similar method to general linear combinations, but with the additional condition that the scalars must be nonnegative - meaning they can either be positive or zero.

For vectors \(u\) and \(v\), a nonnegative linear combination is expressed as \(w = a \cdot u + b \cdot v\) with \(a \geq 0\) and \(b \geq 0\). This results in a comprehensive set of new vectors that essentially create a conical region extending from the origin through the vectors \(u\) and \(v\).

This constraint modifies the general idea of a linear combination by focusing only on those vectors in the first quadrant of a 2D space provided \(u\) and \(v\) aren't collinear or zero. These concepts are heavily utilized in fields like optimization and economics, where ensuring positivity can be crucial to realistic solution modeling.