REVIEW

• From Lesson 5 and Lesson 6, if it's in a blue box, memorize it. Be familiar with all examples.
• From Lesson 6, practice with Subspace Properties (like Question 1, Question 2, and homework problems.)
• From Lesson 7, memorize the blue box containing the definition of Span. Theorems 1, 2, and 3 will not be asked directly, but are used frequently. Be familiar with all examples.
• From Lesson 8, memorize the blue boxes containing the definitions of linearly dependent and linearly independent, the definition of basis, and the definition of dimension. Be familiar with all examples.
• Memorize the standard bases and dimensions given below.
• The basic breakdown of the exam is as follows:
  • Series Solutions (Braun 2.8)
    Find both solutions to an ODE, find unique solution to IVP WITHOUT solving the ODE, know when the solution is a polynomial versus a series (see homework in Braun).
    Remember to simplify all coefficients (as fractions), series solutions must end with $+ \cdots$, the recursion relation must take the form I gave in the online example.
  • Systems (Braun 3.1)
    Focus on example 1 and homework number 5. In particular, we know how to solve systems that can be converted to a collection of $n^{th}$ order ODEs.
  • Linear Dependence (Lesson 8)
    You will be asked to show a set of vectors is linearly dependent using parametric form. Follow the examples I did in class.
  • Subspaces (Lesson 6)
    You may be asked to prove or disprove a Subspace Property. See the notes above.
  • Span Type 1 (Lesson 7)
    You will be asked to show that a vector is in the span of a set. You may be asked for the system of equations (nonhomogeneous), the augmented matrix, or its RREF. Follow Homework from Beezer in Section-S, Exercises S.C15, S.C16, S.C17, S.C20, and S.C21.
  • Span Type 2 (Lesson 7)
    You may be asked to find the spanning set of a subspace. You should follow the process in the homework for Lesson 8, Type 2. I may ask for the homogeneous equation(s) that give you the spanning set, as discussed in class. Note: If you follow the methods in the lesson and like we did in class, your spanning set will actually be a basis, and this is what I am looking for. In particular, just guessing vectors that might span the subspace will not be sufficient.
  • General Info (Lessons 5-8)
    You may be asked a series of short questions testing your general knowledge about linear algebra. See notes above.

• $n$-space, $\mathbb{R}^n$
  The standard basis of $\mathbb{R}^n$ is $\{e_1, e_2, \dots, e_n \}$ and $\textrm{dim}(\mathbb{R}^n) = n$
• Matrices:
  
  • The standard basis of $M_{2 \times 3}(\mathbb{R})$ is $$\left\{ \ \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} \right] , \ \left[ \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 0 \\ \end{array} \right] , \ \left[ \begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{array} \right] , \ \left[ \begin{array}{ccc} 0 & 0 & 0 \\ 1 & 0 & 0 \\ \end{array} \right] , \ \left[ \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 1 & 0 \\ \end{array} \right] , \ \left[ \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] \ \right\}$$ and $\textrm{dim}(M_{2 \times 3}(\mathbb{R})) = 6.$
  • The standard basis of $M_{2}(\mathbb{R})$ is $$\left\{ \ \left[ \begin{array}{cc} 1 & 0 \\ 0 & 0 \\ \end{array} \right] , \ \left[ \begin{array}{cc} 0 & 1 \\ 0 & 0 \\ \end{array} \right] , \ \left[ \begin{array}{cc} 0 & 0 \\ 1 & 0 \\ \end{array} \right] , \ \left[ \begin{array}{cc} 0 & 0 \\ 0 & 1 \\ \end{array} \right] \ \right\}$$ and $\textrm{dim}(M_{2}(\mathbb{R})) = 4.$
  • The standard basis of $D_3$ is $$\left\{ \ \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} \right] , \ \left[ \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ \end{array} \right] , \ \left[ \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \\ \end{array} \right] \ \right\}$$ and $\textrm{dim}(D_3) = 3.$
  • The standard basis of $UT_2$ is $$\left\{ \ \left[ \begin{array}{cc} 1 & 0 \\ 0 & 0 \\ \end{array} \right] , \ \left[ \begin{array}{cc} 0 & 1 \\ 0 & 0 \\ \end{array} \right] , \ \left[ \begin{array}{cc} 0 & 0 \\ 0 & 1 \\ \end{array} \right] \ \right\}$$ and $\textrm{dim}(UT_2) = 3.$
• Polynomials:
  The standard basis of $\mathcal{P}_n$ is $$\{1, t, t^2, \dots, t^n\}$$ and $\textrm{dim}(\mathcal{P}_n) = n+1.$
  (Remember, order matters.)
• Complex Numbers, $\mathbb{C}:$
  The standard basis of $\mathbb{C}$ is $$ \{ 1 , i \}$$ and $\textrm{dim}(\mathbb{C}) = 2$.
• ODE's
  If $L[y]=0$ is a homogeneous ODE of order $n$, then $\textrm{dim}(V_L) = n,$ although we do not yet know how to define the standard basis of $V_L$. For now, we know that any pair of fundamental solutions to a second order, homogeneous ODE forms a basis for $V_L.$ For example, both $$ \{ e^t, e^{-t} \} \quad \textrm{ and } \quad \{ \cosh (t) , \sinh(t) \}$$ are bases of $V_L$ when $L = y^{\ \prime \prime} - y$.
• Systems
  If $A \in M_n(\mathbb{R})$, then $\textrm{dim}(V_A) = n.$
• Kernels
  If $A \in M_{r \times c}(\mathbb{R})$, we will define the standard basis of the kernel of $A,$ $\textrm{ker}(A) = \textrm{Nul}(A),$ as follows:
  Put $A$ in RREF, then put the solutions in paramatric form. Then you will have the general solution in the following form: $$\textrm{Nul}(A) = \left\{ \ r_1 \vec{v}_1 + \cdots + r_k \vec{v}_k \ \middle| \ r_1, \dots, r_k \in \mathbb{R} \ \right\}$$ We define the standard basis of $\textrm{Nul}(A)$ to be $$\left\{ \vec{v}_1 , \dots , \vec{v}_k \right\}$$ and then $\textrm{dim}(\textrm{Nul}(A)) = k.$
  For example, if $$\textrm{RREF}(A) = \left[ \begin{array}{rrr} 1 & -1 & 3 \\ 0 & 0 & 0 \\ \end{array} \right]$$ then the standard basis of $\textrm{Nul}(A)$ is $$\left\{ \ \left[ \begin{array}{r} 1 \\ 1 \\ 0 \\ \end{array} \right] , \left[ \begin{array}{r} -3 \\ 0 \\ 1 \\ \end{array} \right] \ \right\}$$ and $\textrm{dim}(\textrm{Nul}(A)) = 2.$