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To check which options form the same basis, we need to determine if the pairs of functions listed are linearly dependent or independent. Specifically, functions that form the same basis must span the same vector space (they must be linearly independent but span the same space).

### Pairs of functions:
1. \( e^{-t}, e^{8t} \)
2. \( e^t, e^{-8t} \)
3. \( e^{-2t}, e^{4t} \)
4. \( e^{2t}, e^{-4t} \)
5. \( \cos(t), \sin(8t) \)
6. \( \cos(2t), \sin(4t) \)

### Step-by-step reasoning:
For functions involving exponentials, a general basis for functions of the form \( e^{kt} \) involves checking if the exponents \( k \) are distinct or related by some factor, which would affect whether they form the same span.

- **Option 1: \( e^{-t}, e^{8t} \)**  
  These are exponentials with exponents \( -1 \) and \( 8 \). These functions are linearly independent and form a basis for a two-dimensional space of exponential functions.

- **Option 2: \( e^t, e^{-8t} \)**  
  Here, the exponents are \( 1 \) and \( -8 \), which again are distinct. Thus, this pair is also linearly independent and forms a basis for a two-dimensional space of exponential functions.

- **Option 3: \( e^{-2t}, e^{4t} \)**  
  The exponents are \( -2 \) and \( 4 \), distinct and linearly independent. So, this pair forms a basis for a two-dimensional space.

- **Option 4: \( e^{2t}, e^{-4t} \)**  
  With exponents \( 2 \) and \( -4 \), this pair is linearly independent, forming a basis for a two-dimensional space.

Now, comparing the exponents for options 1 to 4, we see that each pair has different distinct exponents, meaning none of them span the same space. They all form different bases.

- **Option 5: \( \cos(t), \sin(8t) \)**  
  This pair includes a cosine and a sine function with different arguments. They are linearly independent, but they do not form a standard trigonometric basis because their frequencies are different.

- **Option 6: \( \cos(2t), \sin(4t) \)**  
  Similar to option 5, these functions have different frequencies and are not part of the same standard basis.

### Conclusion:
None of the pairs form the same basis because the exponents or frequencies are distinct in each case, leading to different spaces being spanned.

Would you like more details on linear independence or function bases?

Here are 5 questions you might find interesting:
1. How can you determine if two functions form a basis for a given space?
2. What is the general method for checking linear independence between functions?
3. How do exponential functions with different exponents relate to vector spaces?
4. Why do trigonometric functions with different frequencies not form the same basis?
5. Can functions of different types (exponentials and trigonometric) ever form the same basis?

**Tip:** In functional analysis, two functions form a basis if they are linearly independent and span the same space—check their structural forms like exponents or frequencies to assess this.

Explore which pairs of functions, including exponential and trigonometric forms, form the same basis. This problem delves into the concepts of linear independence, function spaces, and the characteristics of different function pairs. Suitable for students studying linear algebra at the undergraduate level.

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