Math 311-501 — Test II Review

General Information

Topics Covered

Vector Spaces

• Standard vector spaces. For each these, know what the space is what the operations of vector addition (+) and scalar multiplication (·) corresponding to it.
        Rⁿ,   M_m,n,   P_n,   C[a,b],   C^(k)[a,b],   span{v₁, v₂, ..., vn} .
  
• Subspaces
  • Be able to determine whether a subset S of a vector space V is a subspace. Here is the test: (i) Is 0 in S? (ii) Is S closed under + ? (iii) Is S closed under · ?
  • 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 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 a linear function L:V->W
    • Domain of L, V.
    • Range of L, W
    • Null space of L, N(L) = {v in V | L[v] = 0_W}
    • Image of L, Image(L) = {w in W | w = L[v] for some v in 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, then S is LD.
• Matrix test for vectors in Rⁿ. Let S = {a₁, a₂, ..., a_k} be vectors in Rⁿ. Let A be an n×k matrix with the a_j's as columns.
  • S is LI if and only if N(A) = {0}. Equivalently, S is LD if and only if N(A) contains a nonzero vector.
  • Let k = n. S is LI if and only if A is invertible. Equivalently, S is LD if and only A is singular. In terms of determinants, S is LI if det(A) is not 0, and S is LD if det(A) = 0.
• Basis and Coordinates
  • 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.
  • Matrix subspaces. Be able to find bases for these. See my notes, Methods for Finding Bases.
  • Coordinates. If B = {v₁, v₂, ..., v_n} is a basis for V. then there is a 1:1 correspondence between V and Rⁿ,
    v = x₁v₁ + x₂v₂ + ... + x_nv_n   <-->   [v]_B = (x₁, x₂, ..., x_n)^T .
    The vector [v]_B is called the coordinate vector for v relative to B. Coordinate vectors preserve sums, multiples, and general linear combinations of vectors:
    [u + v]_B = [u]_B + [v]_B ,   [cu]_B = c[u]_B ,   [c₁u₁ + c₂u₂ + ... + c_mu_m]_B = c₁[u₁]_B + c₂[u₂]_B + ... + c_m[u_m]_B
    
• Change of coordinates.
  • Vectors. Let V be a vector space with bases B = {v₁, v₂, ..., vn} and C = {w₁ w₂, ..., wn}. The transition matrix
    S_B->C = [ [v₁]_C [v₂]_C ... [vn]_c ] = [B basis in C coordinates]
    has as columns the C coordinate vectors for the B basis vectors. Know that
    [v]_C = S_B->C[v]_B.
    In addition, S_C->B = (S_B->C)^-1
  • Matrices. If A_B and A_C are the matrices representing L relative to the bases B and C, then
    A_C = S_B->CA_BS_C->B
    This is usually simplified by dropping subscripts. So, let S = S_C->B. Then,
    A_C = S^-1A_BS
    The last equation is called a similarity transformation.
• Dimension
  • 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 arbitrarily large LI sets in it, dim(V) is infinity.
  • Dimensions of matrix subspaces. (See Methods for Finding Bases). The dimension of the column space of an m×n matrix A is called rank(A). This is also the dimension of the row space of A. The dimension of the null space of A is called nullity(A).
  • Important theorems.
    • If S = {v₁, v₂, ..., v_n} spans V, then there is a subset of S that is a basis for V.
    • Let V be finite dimensional. If S = {v₁, v₂, ..., v_n} is LI but does not span V, then we can add vectors to S to make it a basis for V.
    • Let dim(V) = n. S is a basis for V if any two of these hold: (i) S has exactly n vectors in it. (ii) S is LI. (iii) V = span(S). (Theorem 5.9, p. 141.)
    • For any matrix A, rank(A)+nullity(A) = # of columns of A

Linear Functions (Transformations/Operators)

• Definition of a linear function. (These are also called linear transformations and linear operators.) Let V, W be vector spaces. We say that a function L:V->W is linear if L is additive ("sum rule"), L(u+v) = L(u) + L(v), and homogeneous ("constant multiplier rule"), L(cu) = cL(u). These are equivalent to L(au+bv) = aL(u) + bL(v).
• Simple properties.
  • L(0_V) = 0_W
  • L(c₁v₁ + c₂v₂ + ... + c_nv_n) = c₁L(v₁) + c₂L(v₂) + ... + c_nL(v_n)
• One-to-one. A linear function L is one-to-one if and only if L(v) = 0 implies that v = 0.
• Inverse of a linear transformation. If L is one-to-one, then L^-1 : Image(L)->V is linear.
• Matrix representation. If L:V->W is linear, and if B = {v₁, v₂, ..., vn} is a basis for V and C = {w₁, w₂, ..., wm} is a basis for W, then the matrix representing L is
  A = [ [L(v₁)]_C, [L(v₂)]_C, ..., [L(vn)]_C ] = [C coords. of L applied to basis B]
  Know how to find matrix representations for linear transformations. Specific examples are matrix representations for rotations, simple differential operators, reflections, etc.
• Bases for subspaces of a linear transformation L. By using the matrix associated with a linear transformation and the correspondence between vectors in V and column vectors, be able to find bases for the null space and image of L.
• The rank of L = dimension of Image(L), and the nullity of L = dimension of the null space of L. As is the case for matrices, rank + nullity = dimension of V.

Eigenvalues and Eigenvectors

• Definition. Let L:V->V be linear. We say that a scalar λ is an eigenvalue of L if and only if there is a vector v ≠ 0 for which L[v] = λv. The vector v is called an eigenvector of λ.
• Matrix formulation. If dim(V) = n and E = {v₁, v₂, ..., vn} is a basis for V, the eigenvalue problem for A, the matrix of L relative to E.
  • Characteristic polynomial. p_A(λ) = det(A-λI). Know that the eigenvalues of A are exactly the roots of p_A.
  • Be able to find the eigenvalues and eigenvectors for a matrix.
  • Be able to determine whether a matrix can be diagonalized. If it can, be able to find the matrix S such that A = SΛS⁻¹, where Λ = diag(&lambda₁,...,&lambda_n)
• Be able to do simple ODE systems and normal-mode (spring) problems.