What is Linearly Dependent Vectors?

S7-SA2-0084

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

Linearly dependent vectors are a set of vectors where at least one vector can be written as a combination of the others. Imagine them as 'related' or 'redundant' vectors, meaning they don't add new unique information to the set.

Simple Example

Quick Example

Think of prices for a 'chai-samosa combo'. If a chai costs Rs. 10 and a samosa costs Rs. 15, then the combo price of Rs. 25 is 'dependent' on the individual prices. It doesn't give new information. If you knew the chai and samosa prices, you could predict the combo price.

Worked Example

Step-by-Step

Let's check if vectors v1 = (1, 2) and v2 = (2, 4) are linearly dependent.

Step 1: We need to see if we can find non-zero numbers 'a' and 'b' such that a*v1 + b*v2 = (0, 0).
---Step 2: Substitute the vectors: a*(1, 2) + b*(2, 4) = (0, 0).
---Step 3: This gives us two equations: a*1 + b*2 = 0 (Equation 1) AND a*2 + b*4 = 0 (Equation 2).
---Step 4: From Equation 1, a = -2b.
---Step 5: Substitute 'a' into Equation 2: (-2b)*2 + b*4 = 0.
---Step 6: This simplifies to -4b + 4b = 0, which means 0 = 0.
---Step 7: Since we can find non-zero values (e.g., if b=1, then a=-2), the vectors are linearly dependent.
Answer: Yes, v1 and v2 are linearly dependent.

Why It Matters

Understanding linearly dependent vectors is super important in AI/ML, where it helps simplify data and build efficient models. In engineering, it's used to design stable structures and control systems. Even in FinTech, it helps analyze market trends and manage risks, making it a valuable skill for future innovators.

Common Mistakes

MISTAKE: Assuming vectors are dependent only if they are identical. | CORRECTION: Vectors are dependent if one is a multiple of another, or a sum/difference of others, not just identical.

MISTAKE: Confusing linear dependence with orthogonality (vectors being perpendicular). | CORRECTION: Linear dependence means vectors lie on the same line or plane; orthogonality means they are at 90 degrees to each other. These are very different concepts.

MISTAKE: Only checking for two vectors. | CORRECTION: For three or more vectors, you must check if ANY one vector can be written as a combination of the OTHERS, not just one specific pair.

Practice Questions

Try It Yourself

QUESTION: Are the vectors v1 = (3, 0) and v2 = (6, 0) linearly dependent? | ANSWER: Yes

QUESTION: Are the vectors v1 = (1, 0) and v2 = (0, 1) linearly dependent? | ANSWER: No

QUESTION: For vectors u=(1, 1, 0), v=(0, 1, 1), w=(1, 2, 1), check if they are linearly dependent. | ANSWER: Yes (because w = u + v)

MCQ

Quick Quiz

Which of the following describes linearly dependent vectors?

Vectors that are all perpendicular to each other.

Vectors where one vector can be formed by adding or scaling the other vectors.

Vectors that always point in the same direction.

Vectors that have different magnitudes.

The Correct Answer Is:

Option B correctly defines linearly dependent vectors as those where at least one vector can be expressed as a linear combination of the others. Options A, C, and D describe other vector properties but not linear dependence.

Real World Connection

In the Real World

Imagine a drone delivery system like those used by Zepto or Swiggy Instamart. If the drone's movement commands (vectors) include redundant instructions, it wastes energy and time. Identifying linearly dependent command vectors helps optimize the drone's path, making deliveries faster and more efficient, just like ISRO optimizes satellite trajectories.

Key Vocabulary

Key Terms

VECTOR: A quantity with both magnitude and direction, like velocity or force. | SCALAR: A single number that scales a vector, like multiplying a price by 2. | LINEAR COMBINATION: Adding vectors after multiplying them by scalars. | REDUNDANT: Information that is not new or unique, can be derived from existing information.

What's Next

What to Learn Next

Next, you should explore 'Linearly Independent Vectors'. This concept is the opposite of what you just learned and is crucial for understanding how to form a 'basis' for a vector space, which is like building blocks for all other vectors.