Dhanbad

Aastha Sharma replied | 16 hrs ago

A **vector space** is a set of objects called **vectors** along with two operations—**vector addition** and **scalar multiplication**—that satisfy certain rules (axioms).

Formally, a set over a field (like real numbers ℝ) is a vector space if:

1. **Closure under addition:** If , then .

2. **Commutativity of addition:** .

3. **Associativity of addition:** .

4. **Existence of zero vector:** There exists such that .

5. **Existence of additive inverse:** For each , there exists such that .

6. **Closure under scalar multiplication:** If and , then .

7. **Distributive laws:** 

  - 

  - 

8. **Associativity of scalar multiplication:** .

9. **Multiplicative identity:** .

**Example:**

- (all 2D vectors with real components) is a vector space over .

- Polynomials of degree ≤ n form a vector space.

Aastha Sharma replied | 15 hrs ago

A vector space is a set of vectors that can be added together and scaled (multiplied by numbers), while satisfying certain properties like closure, commutativity, associativity, and distributivity.

Sana Begum replied | 5 days ago

create a structured routine by setting specific study times, organizing your workspace, and using a calendar to manage deadlines.

Sonia replied | 20 hrs ago

balance your schedule by using a calendar and planner, communicate with your employer and support network, and create a dedicated study space free from distractions. Stay organized by breaking down tasks, setting clear goals, and leveraging online resources and tools.

Saurabh Verma replied | 24 Oct

Articulation in music means how you play each note — whether it’s smooth, short, or strong.

Here are some simple examples:

Staccato (•) → Play the note short and jumpy.

Legato (curved line) → Play smoothly, join one note to the next.

Accent ( > ) → Play the note louder or stronger.

Tenuto (–) → Hold the note for its full time, a bit gentle.

Marcato ( ^ ) → Play the note very strong and bold.

🎹 In short way:

Articulation shows how to touch or press the keys — softly, sharply, or smoothly — to make the music sound more expressive.

Saurabh Verma replied | 24 Oct

Articulation in music means how you play each note — whether it’s smooth, short, or strong.

Here are some simple examples:

Staccato (•) → Play the note short and jumpy.

Legato (curved line) → Play smoothly, join one note to the next.

Accent ( > ) → Play the note louder or stronger.

Tenuto (–) → Hold the note for its full time, a bit gentle.

Marcato ( ^ ) → Play the note very strong and bold.

🎹 In short way:
Articulation shows how to touch or press the keys — softly, sharply, or smoothly — to make the music sound more expressive.