>GPT-4 with code interpreter is better at math than elite Math undergrads.
Yeah, and I just spent $20 to call BS on what you said.
GPT-4 just bullshits with confidence. I don't see how code interpeter can possibly help here.
Behold:
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Me: I drew three arrows on a piece of paper starting at the same point, pointing left, up, and diagonally (up and right). Do they represent linearly independent vectors?
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ChatGPT: In this case, you have described three different arrows pointing in unique directions:
One pointing left, which can be represented as a vector, say, [-1,0] in a two-dimensional space (using standard Cartesian coordinates).
One pointing up, which can be represented as a vector, say, [0, 1].
One pointing diagonally up and to the right, which could be represented as a vector, say, [1,1].
For these vectors to be linearly independent, no vector can be expressed as a linear combination of the others. In other words, there's no way to add or subtract scaled versions of any two vectors to get the third.
Looking at the vectors you've described:
The first vector represents movement purely in the horizontal direction.
The second vector represents movement purely in the vertical direction.
The third vector represents movement in both directions simultaneously.
Because none of these vectors can be created by scaling and adding the other two (for example, you can't add or subtract any combination of the first and second vectors to get the third one), they are linearly independent.
In mathematical terms, if we form a matrix with these vectors as columns, the determinant of this matrix would not be zero (a non-zero determinant indicates linear independence for the columns of the matrix). This is a more formal way to show their independence, but based on their directions and the explanation above, these vectors are indeed linearly independent.