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Show previous answers 
Aastha Sharma replied | 12 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 | 16 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.
Show previous answers
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.
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Aastha Sharma replied | 12 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.