Math 311-505 - Test II Review

General Information

Topics Covered

Vector Spaces

• Basic notation. Notation for special spaces: Rⁿ, R^m×n, P_n, C[a,b], C^k[a,b].
• Subspaces
  • Test for a subspace. Know the test to determine whether a subset S of a vector space V is a subspace: (i) Is 0 in S? (ii) Is S closed under + ? (iii) Is S closed under · ? Be able to use it to determine whether subsets are subspaces.
  • Span(v₁, v₂, ..., vn) = {All linear combinations of the v_j's}.
  • C^k is a subspace of C^k-1.
  • Subspaces associated with matrices
    • Null space of a matrix A, N(A) = {x in Rⁿ | Ax = 0}.
    • Column space of a matrix. This is the set of all y such that y = Ax; it is the span of the columns of A, Span(a₁, a₂, ..., an).
    • Row space of a matrix. This is the span of all of the rows of A, Span(r₁, r₂, ..., rm).
  • Subspaces associated with linear functions (see chapter 4).
    • Kernel (or null space) of a linear transformation L:V->W,
      ker(L) = {v in V | L[v] = 0_W}.
    • Image of a linear transformation L:V->W for a subspace S of V,
      L(S) = {w in W | w = L[v] for some v in S}.
    • Range of a linear transformation L:V->W, the image of V, L(V).

• Linear Independence and Linear Dependence
  • Definition and test for LI and LD sets of vectors. To test to determine whether a set S of vectors is LI or LD, where S = {v₁, v₂, ..., v_k} , start with the equation
    (*) c₁v₁ + c₂v₂, ..., c_kv_k = 0
    
    • IF the only scalars for which the equation (*) hold are c₁ = c₂ = ... = c_k = 0, then S is LI.
    • IF there are nonzero scalars for which (*) holds are, then S is LD.
  • Matrix test for vectors in Rⁿ. Let S = {x₁, x₂, ..., x_k} be vectors in Rⁿ. Let X be an n×k matrix with the x_j's as columns.
    • S is LI if and only if N(X) = {0}. Equivalently, S is LD if and only if N(X) contains a nonzero vector.
    • Let k = n. S is LI if and only if X is invertible. Equivalently, S is LD if and only X is singular. In terms of determinants, S is LI if det(X) is not 0, and S is LD if det(X) = 0.
  • Wronskian test for LI. IF W[f₁, f₂, ... , f_n](x) is not for one x, then the functions are LI. Test fails if W is identically 0.
  • Representation Theorem. (Be able to show this.)
    Let S = {v₁, v₂, ..., v_n} be an LI subset of a vector space V. IF v belongs to Span(S), then v can be written in one and only one way as a linear combination of vectors in S. That is, the scalars c₁, c₂, ... c_n in the equation below below are unique.
    v = c₁v₁ + c₂v₂ + ... + c_nv_n
    

• Basis and Dimension
  • Definition of basis. A set S = {v₁, v₂, ..., v_n} is a basis for a vector space V if and only if (i) S is LI and (ii) S spans V.
  • Any two bases for a vector space have the same number of vectors in them.
  • Definition of dimension. If V has a basis with n>0 vectors in it, then dim(V) = n. If V ={0}, dim(V) = 0. If V has LI arbitrarily large LI sets in it, dim(V) is infinity.
  • Know Theorem 3.4.3 and 3.4.4 in section 3.4 and be able to do problems using them. Also, know the standard bases for P_n, Rⁿ, R^m×n
  • Know how to find bases for the subspaces associated with a matrix. See my notes, Methods for finding bases.

• Coordinates for Finite Dimensional Vector Spaces
  • Coordinate vectors. Let E = [v₁, v₂, ..., v_n] be an ordered basis for a vector space V. There is a one to one correspondence between vectors in V and column vectors in Rⁿ given by
    v = c₁v₁ + c₂v₂ + ... + c_nv_n   <-->   [v]_E = (c₁, c₂, ..., c_n)^T
    The vector [v]_E is called the coordinate vector for v relative to the basis E.
  • Properties of coordinate vectors. Coordinate vectors preserve sums and multiples of vectors in V. Specifically, they satisfy
    u + v   <-->   [u]_E + [v]_E
    cu   <-->   c[u]_E
    That is,
    [u + v]_E = [u]_E + [v]_E
    [cu]_E = c[u]_E.
    

• Change of Bases (coordinates).

  Let E = [v₁, v₂, ..., v_n], F = [w₁, w₂, ..., w_n] and G = [u₁, u₂, ..., u_n]be bases for V.

  • The transition matrix taking E-coordinates to F-coordinates is
    S_{E -> F} = [ [v₁]_F ... [v_n]_F ]
    In words, the columns of the transition matrix are the basis vectors from E expressed in F coordinates. This matrix does the following
    S_{E -> F} [v]_E = [v]_F
    
  • The transition matrix taking F-coordinates to E-coordinates is the matrix S_{F -> E}, which has columns with the F basis vectors express in terms of E coordinates. One has
    S_{F -> E} = (S_{E -> F})^-1
    
  • A third type of change of basis problem has both E and F vectors expressed relative to a third basis G. Here, one has
    S_{E -> G} [v]_E = [v]_G = S_{F -> G} [v]_F
    To get the transition matrix from E to F, we find
    S_{E -> F} = (S_{F -> G})^-1 S_{E -> G}

• Row and Column Spaces
  • Know what the null space, row space, and column space are for a matrix and be able to find bases for them.
  • Know what the rank and nullity of a matrix are.
  • Know and be able to use this theorem: Rank(A) + Nullity(A) = # of columns of A.
  • Know and be able to apply these consistency theorems. Theorems 3.6.2, 3.6.3, and Corollary 3.6.4.

Linear Transformations

• Definition and Examples
  • Know the definition of a linear transformation. Be able to determine whether a transformation is linear.
  • Subspaces associated with a linear transformation: kernel (or null space), image of a subspace S, L(S), and the range, L(V). Be able to find these space for examples similar to ones in the text.
  • Simple properties. (See p. 193). Identity transformation.
• Matrix Representations
  • Know how to find matrix representations for linear transformations. Be able to work problems similar to ones done in class and to ones in the homework. Specific examples are matrix representations for rotations, simple differential operators, etc.
• Similarity
  • Know the definition of similar matrices.
  • Be able to show that for a linear transformation L:V->V, the matrices representing L relative to different bases are similar.