the basics of linear algebra

What is the dot product of two orthogonal vectors?

The dot product of two orthogonal vectors is zero.
This is because orthogonal means perpendicular, and perpendicular vectors have no overlap in direction, so their dot product is always 0.

Do AA^T and A^TA have the same eigenvalues for any matrix A?

No, AAT and ATA only share the same nonzero eigenvalues (with the same multiplicities).
They can have different numbers of zero eigenvalues because their sizes are different:

• If A is m×n, AAT is m×m and ATA is n×n.

• The extra dimensions in the bigger matrix are filled with zero eigenvalues.
  Only the nonzero eigenvalues are guaranteed to be the same for both matrices.

If a linear transformation from one vector space to another has a nontrivial (nonzero) null space, what does the rank-nullity theorem say about its image?

If the null space is nontrivial, the transformation cannot cover the whole output space (the image is smaller than the full dimension).
Key: A nonzero null space always means the transformation "misses" some directions in the output.

If a square matrix has zero as an eigenvalue, what does the rank-nullity theorem say about its null space?
Can a matrix have full rank and zero as an eigenvalue?

• Zero as an eigenvalue means the matrix is singular, so its null space is nontrivial (nullity > 0 by rank-nullity theorem).

• A matrix with full rank cannot have zero as an eigenvalue—full rank means the null space is trivial (only the zero vector), so all eigenvalues are nonzero.