Linear Independence of t, e^t, sin(t) in Vector Space V

In summary, the conversation discusses the vector space V of all real-valued continuous functions and whether the functions t, e^t, and sin(t) are linearly independent in V. The answer is yes, but the method of proving it is not explicitly stated. The speaker suggests showing the definition of linear independence as the required proof.
  • #1
hkus10
50
0
Let V be the vector space of all real-valued continuous functions.
t, e^t, sin(t) are in V.
Is t, e^t, sin(t) in V linearly independent?
My answer is yes.
However, how can I prove it which is that which do I have to show or can I just say the def of linear independent?
 
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  • #2
hkus10 said:
Let V be the vector space of all real-valued continuous functions.
t, e^t, sin(t) are in V.
Is t, e^t, sin(t) in V linearly independent?
My answer is yes.
However, how can I prove it which is that which do I have to show or can I just say the def of linear independent?

You CAN just say 'def of linear independence'. But that would be a stupid thing to say. What IS the 'def of linear independence'? That's what you have to show.
 
Last edited:

Related to Linear Independence of t, e^t, sin(t) in Vector Space V

1. Is t, e^t, sin(t) linearly independent in Vector Space V?

Yes, t, e^t, sin(t) are linearly independent in Vector Space V. This means that no combination of these vectors can be equal to the zero vector unless all the coefficients are zero.

2. How can we prove the linear independence of t, e^t, sin(t) in Vector Space V?

To prove linear independence, we need to show that the only solution to the equation a(t) + be^t + csin(t) = 0, where a, b, and c are constants, is when a = b = c = 0. This can be done through various methods such as using the Wronskian or showing that the vectors are orthogonal.

3. Can we replace sin(t) with cos(t) and still have linear independence in Vector Space V?

Yes, cos(t) can also be used in place of sin(t) and the set of vectors t, e^t, cos(t) will still be linearly independent in Vector Space V. This is because sin(t) and cos(t) are linearly independent of each other.

4. Is it possible to have more than three vectors that are linearly independent in Vector Space V?

Yes, it is possible to have more than three vectors that are linearly independent in Vector Space V. In fact, any set of n linearly independent vectors in an n-dimensional vector space is considered a basis for that space.

5. How does the linear independence of t, e^t, sin(t) relate to the concept of a basis in Vector Space V?

The linear independence of t, e^t, sin(t) means that these vectors can be used as a basis for Vector Space V. This means that any vector in V can be written as a linear combination of these three basis vectors. It also means that these three vectors span the entire Vector Space V.

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