Question

Find \(a \mathbf{X}+b \mathbf{Y}\) (a) \(\mathbf{X}=(1,2,-3,1), \quad \mathbf{Y}=(0,-1,2,0), a=3, b=6\) (b) \(\mathbf{X}=(1,-1,2), \quad \mathbf{Y}=(0,-1,3), a=-1, b=2\) (c) \(\mathbf{X}=\left(\frac{1}{2}, \frac{3}{2}, \frac{1}{4}, \frac{1}{6}\right), \quad \mathbf{Y}=\left(-\frac{1}{2}, 1,5, \frac{1}{3}\right), a=\frac{1}{2}, b=\frac{1}{6}\)

Short answer

Question: Find the linear combination of the given vectors and coefficients. (a) Vectors: (1, 2, -3, 1) and (0, -1, 2, 0), Coefficients: 3 and 6 (b) Vectors: (1, -1, 2) and (0, -1, 3), Coefficients: -1 and 2 (c) Vectors: (1/2, 3/2, 1/4, 1/6) and (-1/2, 1, 5, 1/3), Coefficients: 1/2 and 1/6 Answer: (a) (3, 0, 3, 3) (b) (-1, -1, 4) (c) (1/6, 5/6, 11/8, 7/36)

Step-by-step solution

Multiply \(\mathbf{X}\) by \(a\)

$$a \mathbf{X} = 3(1, 2, -3, 1) = (3, 6, -9, 3)$$

Multiply \(\mathbf{Y}\) by \(b\)

$$b \mathbf{Y} = 6(0, -1, 2, 0) = (0, -6, 12, 0)$$

Add the two resulting vectors

$$a \mathbf{X} + b \mathbf{Y} = (3, 6, -9, 3) + (0, -6, 12, 0) = (3, 0, 3, 3)$$ #(b) Case (1,-1,2), (0,-1,3), -1, and 2#

Multiply \(\mathbf{X}\) by \(a\)

$$a \mathbf{X} = -1(1, -1, 2) = (-1, 1, -2)$$

Multiply \(\mathbf{Y}\) by \(b\)

$$b \mathbf{Y} = 2(0, -1, 3) = (0, -2, 6)$$

Add the two resulting vectors

$$a \mathbf{X} + b \mathbf{Y} = (-1, 1, -2) + (0, -2, 6) = (-1, -1, 4)$$ #(c) Case \((\frac{1}{2}, \frac{3}{2}, \frac{1}{4}, \frac{1}{6})\), \((-\frac{1}{2}, 1,5, \frac{1}{3})\), \(\frac{1}{2}\), and \(\frac{1}{6}\)

Multiply \(\mathbf{X}\) by \(a\)

$$a \mathbf{X} = \frac{1}{2}\left(\frac{1}{2}, \frac{3}{2}, \frac{1}{4}, \frac{1}{6}\right) = \left(\frac{1}{4}, \frac{3}{4}, \frac{1}{8}, \frac{1}{12}\right)$$

Multiply \(\mathbf{Y}\) by \(b\)

$$b \mathbf{Y} = \frac{1}{6}\left(-\frac{1}{2}, 1, 5, \frac{1}{3}\right) = \left(-\frac{1}{12}, \frac{1}{6}, \frac{5}{6}, \frac{1}{18}\right)$$

Add the two resulting vectors

$$a \mathbf{X} + b \mathbf{Y} = \left(\frac{1}{4}, \frac{3}{4}, \frac{1}{8}, \frac{1}{12}\right) + \left(-\frac{1}{12}, \frac{1}{6}, \frac{5}{6}, \frac{1}{18}\right) = \left(\frac{1}{6}, \frac{5}{6}, \frac{11}{8}, \frac{7}{36}\right)$$

Key concepts

Linear Algebra

Linear algebra is a foundational branch of mathematics that deals with vectors, vector spaces (also known as linear spaces), and linear transformations between these spaces. It is essential in various scientific and engineering disciplines, allowing for the formalization and solution of problems involving linear equations and vectors.

In a typical linear algebra problem, such as the vector addition exercise provided, the core elements include entities known as vectors and matrices. A vector, in this context, is a mathematical object encapsulated by a sequence of numbers, which are its components. These numbers could represent quantities with direction and magnitude in physics or simply elements of a sequence in a more abstract mathematical sense.

Vectors can be added together and can be multiplied by scalars, which leads to operations such as scalar multiplication and vector addition. These operations obey certain rules and properties that mirror those of regular arithmetic, like commutativity, associativity, and distributivity, making them familiar yet unique in their multi-dimensional scope.

Scalar Multiplication

Scalar multiplication refers to the operation of multiplying each component of a vector by a scalar (a single number). This operation is pivotal in changing the magnitude and possibly the direction of the vector, depending on whether the scalar is positive or negative.

For instance, in the exercise provided, we multiply vector \(\mathbf{X}\) by a scalar \(a\), effectively scaling the vector by that amount. The operation affects each component of the vector individually: \(a \mathbf{X} = a(x_1, x_2, ..., x_n) = (ax_1, ax_2, ..., ax_n)\), where \(x_1, x_2, ..., x_n\) are the components of vector \mathbf{X}\, and \(a\) is the scalar. This fundamental concept allows for greater flexibility and adaptability in various vector-related calculations, particularly when combined with addition as seen in the provided exercises.

Vector Operations

Vector operations include a variety of computations such as addition, subtraction, scalar multiplication, and dot product. These are operations that can be performed with vectors, each having specific rules and resulting in either vectors or scalars.

Vector addition, which is showcased in the given exercise, is performed by adding corresponding components of two vectors together. If we consider two vectors \(\mathbf{A} = (a_1, a_2, ..., a_n)\) and \(\mathbf{B} = (b_1, b_2, ..., b_n)\), their sum is obtained by adding each component: \(\mathbf{A} + \mathbf{B} = (a_1 + b_1, a_2 + b_2, ..., a_n + b_n)\).

This operation is crucial for many applications across physics, engineering, and computer science. For example, in physics, vector addition can represent the resultant force when two or more forces are acting on an object. In the context of the given problem, vector addition enables us to combine the effects of scaling two vectors by different scalars and then adding them together to find a resultant vector.