308F/G Winter 2013

• Quiz 1 Problems for Friday January 18: ( Average: ~56% )
  • 3.1: 1-8, 13-17 (Also determine which are subspaces of $\mathbb{R}^{2}$), 22-30,
  • 3.2: 9-17, 29-31,
  • 1.1: 1-6, 19-21 (supposing that these are actually augmented matrices, write out the corresponding systems);
  • 1.2: 1-35, 44-46.
  • Answers to problems given in class.
• Quiz 2 Problems for Friday January 25:( Average: ~%60 )
  • 3.3: 1-7 and 12-19 (just give the algebraic descriptions, e.g. $W=\{x:x_{1}=x_{2}\}$), 20,21
  • 3.4: 1-10, 21-24(a), 30 (don't use the hint, use this one instead: you can show that they span all of $\mathbb{R}^{3}$ by showing that their span contains another triple of vectors that also span $\mathbb{R}^{3}$, why is this enough?)
  • Also, here is some help with the basis problems, since I won't be able to cover this until Wednesday.
  • Answers to problems given in class.
• Quiz 3 Problems for Friday February 1:( Average: ~%89 )
  • 3.4: 11-16 (a) and (d), 21-24 (a) and (b), 27 (the overlap with last week's problems is intentional)
  • 3.5: 1-20, 25-26 (don't worry about giving the nullity and rank), 27,30-32.
  • Answers to problems given in class.
• Quiz 4 Problems for Friday February 8: ( Average: ~%85 )
  • 3.6: 1-18.
  • Answers to problems given in class.
• Quiz 5 Problems for Friday February 15 ( Average: ~%75 ) (also, here are some notes about matrix multiplication):
  • 1.5: 1-6, 25-41,53-5, 59, 61,62.
  • 3.7: 1-2,4, 6, 8-17,19,31-37,39,44
  • Answers to problems given in class.
• Quiz 6 Problems for Friday February 22 ( Average: ~%66 )(also, here are some notes about the rank-nullity theorem) and some notes about inverses of matrices and linear transformations..
  • 1.6: 1-6, 27, 28, 43,56-62.
  • 1.9: 9-26,
    
  • 3.5: 21-26
  • 3.7: 3, 5,25-30
  • Extra Problems:
    1. If $W$ is a subspace of a subspace $V$ and $\dim W=\dim V$, show that $W=V$.
    2. If $T:V\rightarrow W$ is a linear transformation between subspaces $V$ and $W$, show that $N(T)$ is a subspace of $V$ and $R(T)$ is a subspace of $W$.
    3. Show that if $T:V\rightarrow W$ is linear and $N(T)=\{0\}$, then whenever $T(v)=T(u)$, we must have $u=v$.
    4. If $v_{1},...,v_{n}$ is a basis for $V$, and $T:V\rightarrow W$ is linear, then $T(v_{1}),...,T(v_{n})$ span $R(T)$. If $N(T)=\{0\}$, then $T(v_{1}),...,T(v_{n})$ is a basis for $R(T)$. Conversely, show that if $v_{1},...,v_{n}$ is a basis for $V$ and $T(v_{1}),...,T(v_{n})$ is a basis for $R(T)$, then $N(T)=\{0\}$.
    5. If $T:V\rightarrow W$ is linear, and $\dim W+nullity(T)=\dim V$, then $R(T)=W$. 
  • Answers to problems given in class.
• Quiz 7 Problems for Friday March 1:
  • 4.2: 8-19, 24, 25,27-30.
• Quiz 8 Problems for Friday March 8:
  • 4.1: 1-12
  • 4.3: 1-10
  • 4.4: 1-12
    
  • 4.5: 1-17,21,22,24
• Recommended Problems from sections 4.6 and 4.7:
  • 4.6: 1-30.
  • 4.7: 1-12, 25-27.

• Exam 1 will be Monday February 4. This will be based on sections 1.1, 1.2, 3.1-3.5. Here is a practice midterm, and here are its solutions . Here are the midterm 1 solutions. ( Average: ~%70 )
• Exam 2 will be Wednesday February 27. This will be based upon sections 1.5,1.6,1.7,1.9, and 3.6, and 3.7 Here is a practice midterm and here are its solutions . (Note, problem 4c should read $(A^{-1}(AB)^2C^{-1}(BA)^{-1}$, which will make the problem a lit easier. In fact, once you observe that $A^{-1}=C$, then $(A^{-1}(AB)^{2} C^{-1}(BA)^{-1}$ = $A^{-1}ABABC^{-1}A^{-1}B^{-1}$ = $BABB^{-1}=BA$. Also, problem 2a should have answer (-1,1,0) instead of (1,1,0).) Here are the midterm 2 solutions. ( Average: ~%66 )
• The final exam is comprehensive, it will cover all material covered on the quizzes plus sections 4.6 and 4.7. Please check my.uw.edu for the date and time of your final. Here is a practice exam and here are the solutions .

This isn't high school, grow up.