Vectors: The Foundation of Linear Algebra

Vectors are foundational elements of linear algebra . They are essential in mathematics, particularly for data science. Fundamentally, a vector can be thought of simply as a list .

Basics and Representation

In practical terms, vectors often arise from tabular data . For instance, if you look at a table showing India’s GDP or cricket team batting averages, a vector corresponds to a row or a column of that data.

Row Vector : A list of numbers arranged horizontally, representing a row of data.
- Example: If we take the batting averages of five players against South Africa from a table, we get a row vector like $(64.35, 31.92, 33.30, 26.00, 49.87)$ .
Column Vector : A list of numbers arranged vertically, representing a column of data.
- Example: The averages scored by R. Sharma against several teams can be represented as a column vector:
  
  Vertical List
  
  $\begin{pmatrix} 61.33 \\ 50.44 \\ 33.47 \\ 33.30 \\ 46.25 \\ 51.42 \end{pmatrix}$

Vectors in $R^n$ (n-dimensional real space) are lists, rows, or columns containing $n$ real entries, and they are equivalent to points in $R^n$ .

The Importance of Vectors

Vectors are important because they allow for arithmetic operations on these lists.

For example, if you wanted to find the average sectoral GDP over 10 years, you could represent each year’s data as a row vector. You would add these 10 row vectors coordinate-wise and then multiply the resultant vector by the scalar $\frac{1}{10}$ (or $\frac{1}{10}$ times each element) to find the average GDP per sector.

Visual Anchors: Understanding Vectors Geometrically

💡

Visual Anchor: R² Representation

Imagine a standard Cartesian plane (x-axis and y-axis). A key method for visualizing a vector is representing a point $(a, b)$ in $R^2$ as an arrow joining the origin $(0, 0)$ to the point $(a, b)$ .

The vector $(1, 2)$ is visualized by an arrow starting at the origin $(0, 0)$ and ending at the point $(1, 2)$ . Similarly, the vector $(-1, -1)$ is an arrow starting at $(0, 0)$ and ending at $(-1, -1)$ .

A vector in a physical context is defined by two fundamental properties: magnitude (size) and direction. The length of the arrow represents the magnitude, and the arrowhead indicates the direction. Examples of vectors in physics include Velocity, Acceleration, and Force. Importantly, a vector does not have position; it remains the same if displaced parallel to itself without changing its length.

Vector Operations

A. Addition: Vector addition is done coordinate-wise. Example: If Arun’s demands are $(3, 2)$ (3 Kg Rice, 2 Kg Dal) and Neela’s demands are $(5, 6)$ , their total demand vector is $(3, 2) + (5, 6) = (8, 8)$ .

B. Scalar Multiplication: Multiplying a vector by a scalar (a real number) means multiplying each entry of the vector by that scalar. Example: If Buyer A buys the same items two days in a row, the total quantity vector can be found by scalar multiplication: $2 \cdot (8, 8, 4, 10, 4) = (16, 16, 8, 20, 8)$ .

💡

Visual Anchor: Vector Addition

Vector addition in $R^2$ can be visualized using the head-to-tail method or the parallelogram law.

When adding two vectors, say $(1, 2)$ and $(2, 1)$ , draw them both starting from the origin. Complete the figure to form a parallelogram. The diagonal of the parallelogram starting from the origin represents the resultant vector from the addition.

Example: $(1, 2) + (2, 1) = (3, 3)$ . The resultant vector $(3, 3)$ is the diagonal connecting $(0, 0)$ to $(3, 3)$ .

Lesson Questions

We can apply these basic rules to solve problems involving vector arithmetic. Let $A = (1, 2)$ and $B = (2, 3)$ .

Vector Operations

Choose the set of correct options using the vectors $A=(1, 2)$ and $B=(2, 3)$ .

Option 1: $2A$ is the vector $(2, 4)$ .
Option 2: $3B$ is the vector $(6, 9)$ .
Option 3: $A + B$ is the vector $(3, 5)$ .
Option 4: $A - B$ is the vector $(-1, -1)$ .

📝 View Detailed Solution ▼

All options are correct, based on scalar multiplication and coordinate-wise addition/subtraction.

Option 1 (Scalar Multiplication): $2A = 2 \cdot (1, 2) = (2 \cdot 1, 2 \cdot 2) = (2, 4)$ . (Correct)
Option 2 (Scalar Multiplication): $3B = 3 \cdot (2, 3) = (3 \cdot 2, 3 \cdot 3) = (6, 9)$ . (Correct)
Option 3 (Addition): $A + B = (1, 2) + (2, 3) = (1+2, 2+3) = (3, 5)$ . (Correct)
Option 4 (Subtraction): $A - B = (1, 2) - (2, 3) = (1-2, 2-3) = (-1, -1)$ . (Correct)

Finding a Scalar

Let $A = (1, 1, 1)$ and $B = (2, -1, 4)$ be two vectors. Suppose $cA + 3B = (4, j, k)$ , where $c, j, k$ are real numbers (scalars). Find the value of $c$ .

📝 View Detailed Solution ▼

We use scalar multiplication and coordinate-wise addition. $cA + 3B = c(1, 1, 1) + 3(2, -1, 4)$ $cA + 3B = (c, c, c) + (6, -3, 12)$ $cA + 3B = (c + 6, c - 3, c + 12)$

We are given that $cA + 3B = (4, j, k)$ . By equating the first components: $c + 6 = 4$ $c = 4 - 6 = -2$

Result

The value of

c

-2

Finding Unknown Components

Given the result $c=-2$ from Question 2, find the value of $j + k$ .

📝 View Detailed Solution ▼

Substitute $c = -2$ back into the resultant vector derived in Question 2: $(-2)A + 3B = ((-2) + 6, (-2) - 3, (-2) + 12)$ $(-2)A + 3B = (4, -5, 10)$

Since the resultant vector is $(4, j, k)$ , we have $j = -5$ and $k = 10$ .

$j + k = -5 + 10 = 5$

The value of $j + k$ is $5$ .

To summarize, vectors provide an algebraic framework for dealing with multi-dimensional data or physical quantities by treating lists of numbers as single objects that can be scaled (multiplied by a number) or combined (added coordinate-wise), often visualized as arrows in space.