Answer Key: Linear Algebra Course Quiz

1 Answers

1. (r(t)=(1,2)+t(3,-1)=(1+3t,2-t)).
2. It contains (0), and if (Au=0) and (Av=0), then (A(au+bv)=aAu+bAv=0).
3. The inner product is (2+0-3=-1); the norms are () and ().
4. ().
5. ().
6. If (A=[a_1  a_n]), then (Ax=_i x_ia_i), so the columns encode the images of the standard basis vectors.
7. No: ((2,4)=2(1,2)).
8. Solve (a(1,1)+b(2,-1)=(5,1)), giving ().
9. ().
10. ((V)=(A)+(A)); dimensions split into independent output directions and input directions sent to zero.
11. Free variables are parameters; pivot variables are expressed in terms of them, producing infinitely many solutions when at least one free variable remains.
12. ().
13. A square matrix is invertible iff ((A)).
14. (p_A()=(3-)(2-)).
15. For (), (E_3={(1,0)}); for (), (E_2={(-1,1)}).
16. A linear dependence among eigenvectors with distinct eigenvalues, acted on by (A), produces a second relation with distinct scalar factors; subtracting forces all coefficients to vanish.

2 Grading

Award 2 points for a correct and explicit answer, 1 for the correct method with a minor error, and 0 for a missing or materially incorrect answer.