It’s unclear to me why the answer to this question is just first vectors set and not both (?)
If I understand correctly for non-singular 3x3 matrix, any linearly independent three vectors, will span the matrix, and every answer option points a linearly independent vector set.
The only difference is that for second option we miss arrow over the letter, does it indicate it’s not a vector, so we should assume as false?
I think the problem statement should be “Which of the vectors FORM the matrix W? (Not span)”
Below is my humble opinion.
This problem can be a bit puzzling, so let’s break it down. What it seems the teacher wants to check is whether you understand the difference between ‘spanning a vector space’ and ‘forming a matrix’.
Both Option 1 and Option 2 consist of linearly independent vectors, which means they can both span the 3D vector space. This means you could use vectors from either option to form the columns of a 3x3 matrix, which we’ll call ‘W’.
The key difference is this: In Option 1, the vectors directly form the columns of the matrix W, so to represent W using these vectors, the weights (or scalars) you’d multiply each vector by are simply 1. It’s straightforward.
However, with Option 2, to represent W, you’d need to figure out a combination of these vectors, meaning you’d have to calculate specific weights for each vector.
So, while both options can represent W, only Option 1 does so directly with weights of 1 for each vector. This is likely why Option 1 is deemed correct.
It’s a tricky point, and I understand the confusion.
