Example: One-shot / few-shot learning

It鈥檚 often helpful to use something akin to one-shot learning or few-shot learning when engineering prompts. This approach involves providing a concrete example or a few concrete examples to show the model what you鈥檇 like in its response.

First, we鈥檒l provide a na茂ve prompt, show where it goes astray, and then see how we can improve our results by providing a single example.

Say you鈥檙e teaching a first course in linear algebra, and you鈥檇 like some problems either for classroom demonstration or as an assignment.

The na茂ve prompt

Na茂ve prompt

Can you give me three problems that require Gauss-Jordan reduction?

ChatGPT鈥檚 response to the na茂ve prompt

ChatGPT鈥檚 response

Sure! Here are three problems that require Gauss-Jordan elimination for their solutions:

Problem 1: Solving a System of Linear Equations

Solve the following system of equations using Gauss-Jordan elimination:

2x + 3y - z = 5

4x - y + 5z = 6

-2x + 4y + 3z = 7

Problem 2: Finding the Inverse of a Matrix

Find the inverse of the following matrix using Gauss-Jordan elimination:

A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{bmatrix}

(Hint: Augment ( A ) with the identity matrix and row-reduce.)

Problem 3: Determining Linear Dependence of Vectors

Determine whether the following vectors are linearly dependent using Gauss-Jordan elimination:

v_1 = (1, 2, 3, 4), \quad v_2 = (2, 4, 6, 8), \quad v_3 = (3, 6, 9, 12)

(Hint: Form a matrix with these vectors as rows and row-reduce to check for a row of zeros.)

Would you like me to go through the solutions as well?

Commentary on ChatGPT鈥檚 response to the na茂ve prompt

This is interesting. It鈥檚 not incorrect, and for some, this might be exactly the answer they were looking for: what kinds of problems can I solve using Gauss-Jordan elimination?

However, if the instructor is looking for a few elementary examples of solving systems of linear equations, they鈥檇 need some prompt engineering.

Claude鈥檚 response to the na茂ve prompt

Claude鈥檚 response

I鈥檒l provide three problems that can be solved using Gauss-Jordan reduction, arranged in increasing order of complexity.

1. System of Linear Equations: Solve the following system:

2x + y - z = 4
3x - y + 2z = 1
x + y + z = 6

2. Matrix Inverse: Find the inverse of the following matrix, if it exists:

[2  1  0]
[1  3  1]
[0  1  2]

3. Finding Basis for Nullspace: Find a basis for the nullspace of the following matrix:

[1  2  1  -1]
[2  4  0   2]
[3  6  1   1]

For each of these, Gauss-Jordan reduction would be used differently: - For the system of equations, we鈥檇 reduce the augmented matrix to row echelon form - For the matrix inverse, we鈥檇 use the augmented matrix [A|I] and reduce to [I|A鈦宦筣 - For the nullspace basis, we鈥檇 reduce to row echelon form and then use the free variables to find vectors in the nullspace

Would you like me to demonstrate the solution process for any of these problems?

Commentary on Claude鈥檚 response to the na茂ve prompt

Claude鈥檚 response is remarkably similar to that of ChatGPT.

A better prompt

Adding a single example is sufficient to get a much more focused response.

Prompt with single example

Here鈥檚 an example of what I鈥檓 after:

Solve:

x + y - 2z = -2 \\ y + 3z = 7 \\ x - z = -1

Can you solve this, and then produce three more problems of similar difficulty and number of variables?

ChatGPT鈥檚 response

ChatGPT鈥檚 response

The solution to the given system:

x + y - 2z = -2

y + 3z = 7

x - z = -1

is:

x = 1, \quad y = 1, \quad z = 2

---

Here are three more problems of similar difficulty:

Problem 1:

Solve the system using Gauss-Jordan elimination:

x + 2y - z = 3

2x - y + 3z = 4

3x + y + 2z = 5

---

Problem 2:

Solve the system using Gauss-Jordan elimination:

2x - y + 3z = 7

x + 3y - 2z = -4

3x + y + z = 5

---

Problem 3:

Solve the system using Gauss-Jordan elimination:

x - 2y + z = 4

3x + y - 4z = -3

2x - y + 3z = 5

Would you like me to solve these as well?

Claude鈥檚 response

Claude鈥檚 response

I鈥檒l solve this example step by step, then give you three similar problems.

First, let鈥檚 solve the given system using Gauss-Jordan reduction:

The augmented matrix is:

[1  1  -2 | -2]
[0  1   3 |  7]
[1  0  -1 | -1]

Step 1: Subtract row 1 from row 3 to eliminate x

[1  1  -2 | -2]
[0  1   3 |  7]
[0 -1   1 |  1]

Step 2: Add row 2 to row 3 to eliminate y

[1  1  -2 | -2]
[0  1   3 |  7]
[0  0   4 |  8]

Step 3: Solve for z using row 3: z = 2

Step 4: Substitute z = 2 into row 2: y + 3(2) = 7, so y = 1

Step 5: Substitute z = 2 and y = 1 into row 1: x + 1 - 4 = -2, so x = 1

Solution: x = 1, y = 1, z = 2

Here are three similar problems:

1. Solve: x + 2y - z = 3 2y + 3z = 8 x + z = 2

2. Solve: 2x - y + z = 4 y - 2z = -3 x + y = 5

3. Solve: x + y + z = 6 2y - z = 1 x - z = 2

Each of these systems has: - Three variables (x, y, z) - Three equations - Integer coefficients that are small (-3 to 3) - Solutions with reasonable numbers - Similar complexity in the reduction process

Would you like me to solve any of these new problems?

Commentary on responses by ChatGPT and Claude

By providing a single example in the prompt we get more-or-less exactly what we were after.

Claude鈥檚 response includes augmented matrices which is nice but not essential.