Hints

Use these only after trying the exercises.

MATH-C04-C-001

Add matching coordinates.

Compute b - a first. The difference should make a familiar triangle.

Multiply matching entries and add. Orthogonal vectors give zero.

The vector u = [1, 0] keeps the horizontal component.

Compute the dot product first, then add the bias.

Check whether [0,0] satisfies the shifted line equation.

Both proposed basis vectors point in the same direction.

Nearby embeddings can show a relationship under the model's representation, but that does not prove why the model learned that relationship.

Use drawings for intuition and computation for the actual high-dimensional space.

Ask what the operation is doing to the space.

Subtract the projection from the original vector. The residual is what the projection did not keep.

The boundary is the set of points with score zero. Multiplying zero by 2 is still zero.

The coordinates [2, 3] mean 2b_1 + 3b_2, not automatically [2, 3] in the standard axes.