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MATH 240 Spring, 2006
Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 8th Ed. Sections 1.1–1.6, 2.1–2.2, 3.2–3.8, 4.3–4.5, 5.1–5.3, 5.5, 6.1–6.5, 7.1–7.2, 7.
Chapter 1: Linear Equations and Matrices
DEFINITIONS
There are a number of definitions concerning systems of linear equations on pages 1–2 and 26–27. The numbered definitions 1.1–1.4 and 1.6–1.7 concern matrices and matrix operations.
1.5 (p 18). If A = [aij ] is an m × n matrix, then its transpose is AT^ = [aji].
1.8–1.9 (p 40). A square matrix A is symmetric if AT^ = A and skew symmetric if AT^ = −A.
1.10 (p 43). An n × n matrix A is invertible (also called nonsingular) if it has an inverse matrix A−^1 such that AA−^1 = In and A−^1 A = In. If A is not invertible, it is called a singular matrix.
(p 30 Ex #41). The trace of a square matrix is the sum of the entries on its main diagonal.
THEOREMS
1.1–1.4 (pp 32–35) These theorems list the basic properties of matrix operations. The only surprises are that we can have AB 6 = BA, and that (AB)T^ = BT^ AT^.
1.5–1.8 (pp 43–45) Inverses are unique. If A and B are invertible n × n matrices, then AB, A−^1 , and AT^ are invertible, with (AB)−^1 = B−^1 A−^1 , (A−^1 )−^1 = A, and (AT^ )−^1 = (A−^1 )T^.
USEFUL EXERCISES
(p 48 Ex #42). The matrix A =
[
a b c d
]
is invertible if and only if ad − bc 6 = 0, and in this case A−^1 = 1 ad − bc
[
d −b −c a
]
Chapter 2: Solving Linear Systems
DEFINITIONS
2.1 (p 78). The definitions of row echelon form and reduced row echelon form are easier to understand than to write down. You need to know how to use them.
2.2 (p 79) These are the elementary row operations on a matrix: (I) interchange two rows; (II) multiply a row by a nonzero number; (III) add a multiple of one row to another.
2.3 (p 80) Two matrices are row equivalent if one can be obtained from the other by doing a finite sequence of elementary row operations.
2.4 (p 101) A matrix that is obtained from the identity matrix by doing a single elementary row operation is called an elementary matrix.
THEOREMS
2.1–2.2 (pp 80–83). Every matrix is row equivalent to a matrix in row echelon form, and row equivalent to a unique matrix in reduced row echelon form.
2.3 (p 85). Linear systems are equivalent if and only if their augmented matrices are row equivalent.
2.4 (p 95). A homogeneous linear system has a nontrivial solution whenever it has more unknowns than equations.
2.5–2.6 (p 102). Elementary row operations can be done by multiplying by elementary matrices.
2.7 (p 102). Any elementary matrix has an inverse that is an elementary matrix of the same type.
2.8 (p 103). A matrix is invertible if and only if it can be written as a product of elementary matrices.
2.9 (p 102). The homogeneous system Ax = 0 has a nontrivial solution if and only if A is singular.
2.10 (p 106). A square matrix is singular if and only if it is row equivalent to a matrix with a row of zeros.
2.11 (p 108). If A and B are n × n matrices with AB = In, then BA = In and thus B = A−^1.
The summary on page 104 is important:
The following conditions are equivalent for an n × n matrix A: (1) A is invertible; (2) Ax = 0 has only the trivial solution; (3) A is row equivalent to the identity; (4) for every vector b, the system Ax = b has a unique solution; (5) A is a product of elementary matrices.
ALGORITHMS
- (pp 85–91) Gauss-Jordan reduction. This algorithm puts a matrix in reduced row echelon form. You need to know how to it quickly and accurately. Gaussian elimination uses back substitution after putting the matrix in row echelon form.
- (pp 105–106) To find the inverse A−^1 of an invertible matrix A, start with [A|In] and use Gauss-Jordan reduction to reduce it to [In|A−^1 ].
- (p 103) To write an invertible matrix A as a product of elementary matrices, row reduce it to the identity and keep track of the corresponding elementary matrices. If Ek · · · E 2 E 1 A = In, then A = E− 1 1 E 2 − 1 · · · E k− 1.
3.9 (p 171). If a vector space has a basis with n elements, then it cannot contain more than n linearly independent vectors.
Cor (pp 172–173). Any two bases have the same number of elements; dim(V ) is the maximum number of linearly independent vectors in V ; dim(V ) is the minimum number of vectors in any spanning set.
3.10 (p 174). Any linearly independent set can be expanded to a basis.
3.11 (p 175). If dim(V ) = n, and you have a set of n vectors, then to check that it forms a basis you only need to check one of the two conditions (spanning and linear independence).
3.16 (p 202). Row equivalent matrices have the same row space.
3.17 (p 206). For any matrix (of any size), the row rank and column rank are equal.
3.18 [The rank-nullity theorem] (p 207). If A is any m × n matrix, then rank(A) + nullity(A) = n.
3.20 (p 209). An n × n matrix has rank n if and only if it is row equivalent to the identity.
3.20 (p 210). Ax = b has a solution if and only if the augmented matrix has the same rank as A.
Summary of results on invertibility (p 211). The following are equivalent for any n × n matrix A: (1) A is invertible (3) A is row equivalent to the identity (5) A is a product of elementary matrices (4) Ax = b always has a solution (2) Ax = 0 has only the trivial solution (6) rank(A) = n (7) nullity(A) = 0 (8) the rows of A are linearly independent (9) the columns of A are linearly independent
ALGORITHMS
- (p 156). To test whether the vectors v 1 , v 2 ,... , vk are linearly independent or linearly dependent: Solve the equation x 1 v 1 + x 2 v 2 +... + xkvk = 0. Note: the vectors must be columns in a matrix. If the only solution is all zeros, then the vectors are linearly independent. If there is a nonzero solution, then the vectors are linearly dependent.
- (p 154). To check that the vectors v 1 , v 2 ,... , vk span the subspace W : Show that for every vector b in W there is a solution to x 1 v 1 + x 2 v 2 +... + xkvk = b.
- (p 170). To find a basis for the subspace span{v 1 , v 2 ,... , vk} by deleting vectors: (i) Construct the matrix whose columns are the coordinate vectors for the v’s (ii) Row reduce (iii) Keep the vectors whose column contains a leading 1 Note: The advantage here is that the answer consists of some of the vectors in the original set.
- (p 174). To find a basis for a vector space that includes a given set of vectors, expand the set to include all of the standard basis vectors and use the previous algorithm.
- (p 178). To find a basis for the solution space of the system Ax = 0 : (i) Row reduce A (ii) Identify the independent variables in the solution (iii) In turn, let one of these variables be 1, and all others be 0 (iv) The corresponding solution vectors form a basis
- (p 184). To solve Ax = b, find one particular solution vp and add to it all solutions of the homogeneous system Ax = 0.
- (p 193). To find the transition matrix PS←T : Note: The purpose of the procedure is to allow a change of coordinates [v]S = PS←T · [v]T. (i) Construct the matrix A whose columns are the coordinate vectors for the basis S (ii) Construct the matrix B whose columns are the coordinate vectors for the basis T (iii) Row reduce the matrix [ A | B ] to get [ I | PS←T ] Shorthand notation: Row reduce [ S | T ] [ I | PS←T ]
- (p 202) To find a simplified basis for the subspace span{v 1 , v 2 ,... , vk}: (i) Construct the matrix whose rows are the coordinate vectors for the v’s (ii) Row reduce (iii) The nonzero rows form a basis Note: The advantage here is that the vectors have lots of zeros, so they are in a useful form.
Chapter 4: Inner Product Spaces
DEFINITIONS
4.1 (p 235). Let V be a real vector space. An inner product on V is a function that assigns to each ordered pair of vectors u, v in V a real number (u, v) satisfying: (1) (u, u) ≥ 0; (u, u) = 0 if and only if u = 0. (2) (v, u) = (u, v) for any u, v in V. (3) (u + v, w) = (u, w) + (v, w) for any u, v, w in V. (4) (cu, v) = c(u, v) for any u, v in V and any scalar c.
4.2 (p 240). A vector space with an inner product is called an inner product space. It is a Euclidean space if it is also finite dimensional.
Def (p 239). If V is a Euclidean space with ordered basis S, then the matrix C with (v, w) = [v]TS C [w]S is called the matrix of the inner product with respect to S (see Theorem 4.2).
4.3 (p 243). The distance between vectors u and v is d(u, v) = ||u − v||.
4.4 (p 243). The vectors u and v are orthogonal if (u, v) = 0.
4.5 (p 243). A set S of vectors is orthogonal if any two distinct vectors in S are orthogonal. If each vector in S also has length 1, then S is orthonormal.
4.6 (p 260). The orthogonal complement of a subset W is W ⊥^ = {v in V | (v, w) = 0 for all w in V }.
Def (p 267). Let W be a subspace with orthonormal basis w 1 ,... , wm. The orthogonal projection of a vector v on W is projW (v) =
∑m i=1(v,^ wi)wi.
THEOREMS
4.2 (p 237). Let V be a Euclidean space with ordered basis u 1 , u 2 ,... , un}. If C is the matrix [(ui, uj )], then for every v, w in V the inner product is given by (v, w) = [v]TS C [w]S.
4.3 (p 240). If u, v belong to an inner product space, then |(u, v) ≤ ||u|| ||v||. (Cauchy–Schwarz inequality)
Cor (p 242). If u, v belong to an inner product space, then ||u + v|| ≤ ||u|| + ||v||. (Triangle inequality)
4.4 (p 244). A finite orthogonal set of nonzero vectors is linearly independent.
4.5 (p 248). Relative to an orthonormal basis, coordinate vectors can be found by using the inner product.
4.6 (p 249). Any finite dimensional subspace of an inner product space has an orthonormal basis.
4.7 (p 253). If S is an orthonormal basis for an inner product space V , then (u, v) = [u]S · [v]S.
4.9 (p 260). If W is a subspace of V , then W ⊥^ is a subspace with W ∩ W ⊥^ = { 0 }.
4.10 (p 262). If W is a finite dimensional subspace of an inner product space V , then every vector in V can be written uniquely as a sum of a vector in W and a vector in W ⊥. Notation: V = W ⊕ W ⊥
4.12 (p 263). The null space of a matrix is the orthogonal complement of its row space.
ALGORITHMS
- (p 249). To find an orthonormal basis for a finite dimensional subspace W : (Gram–Schmidt process)
(i) Start with any basis S = {u 1 ,... , um} for W (ii) Start constructing an orthogonal basis T ∗^ = {v 1 ,... , vm} for W by letting v 1 = u 1. (iii) To find v 2 , start with u 2 and subtract its projection onto v 1. v 2 = u 2 −
(u 2 , v 1 ) (v 1 , v 1 ) v 1
(iv) To find vi, start with ui and subtract its projection onto the span of the previous vi’s vi = ui −
(ui, v 1 ) (v 1 , v 1 )
v 1 −
(ui, v 2 ) (v 2 , v 2 )
v 2 −... −
(ui, vi− 1 ) (vi− 1 , vi− 1 )
vi− 1
(v) Construct the orthonormal basis T = {w 1 ,... , wm} by dividing each vi by its length wi =
||vi||
vi
Chapter 6: Determinants
Important note: the definitions and theorems in this chapter apply to square matrices.
DEFINITIONS
6.4 (p 373). Let A = [aij ] be a matrix. The cofactor of aij is Aij = (−1)i+j^ det(Mij ), where Mij is the matrix found by deleting the ith row and j th column of A.
6.5 (p 381). The adjoint of a matrix A is adj(A) = [Aji], where Aij is the cofactor of aij.
THEOREMS
Results connected to row reduction:
6.2 (p 363). Interchanging two rows (or columns) of a matrix changes the sign of its determinant.
6.5 (p 365). If every entry in a row (or column) has a factor k, then k can be factored out of the determinant.
6.6 (p 365). Adding a multiple of one row (or column) to another does not change the determinant.
6.7 (p 366). If A is in row echelon form, then det(A) is the product of terms on the main diagonal.
Some consequences: 6.3 (p 364). If two rows (or columns) of a matrix are equal, then its determinant is zero.
6.4 (p 364). If a matrix has a row (or column) of zeros, then its determinant is zero.
Other facts about the determinant:
6.1 (p 363). For the matrix A, we have det(AT^ ) = det(A).
6.8 (p 369). The matrix A is invertible if and only if det(A) 6 = 0.
6.9 (p 369). If A and B are both n × n matrices, then det(AB) = det(A) · det(B).
Cor (p 370). If A is invertible, then det(A−^1 ) =
det(A)
Cor (p 370). Similar matrices have the same determinant.
Expansion by cofactors:
6.10 (p 374). If A is an n × n matrix, then det(A) =
∑n j=1 aij^ Aij^. (expansion by cofactors along row i) Note: This can be used to define the determinant by induction, rather than as in Def 6.2 on p 359.
6.12 (p 382). For the matrix A we have A · adj(A) = det(A) · I and adj(A) · A = det(A) · I.
Cor (p 383). If A is invertible, then A−^1 =
det(A) adj(A).
5.13 [Cramer’s rule] (p 385). If det(A) 6 = 0, then the system Ax = b has solution xi =
det(Ai) det(A) , where Ai is the
matrix obtained form A by replacing the ith column of A by b.
ALGORITHMS
- (p 367). Determinants can be computed via reduction to triangular form. (Keep track of what each elementary operation does to the determinant.)
- (p 374). Determinants can be computed via expansion by cofactors.
- (p 334). To find the area of a parallelogram in R^2 : (i) Find vectors u, v that determine the sides of the parallelogram. (ii) Find the absolute value of the determinant of the 2 × 2 matrix with columns u, v. OR (i) Put the coordinates of 3 vertices into the 3 × 3 matrix given on p 377. (ii) Find the absolute value of the determinant of the matrix.
- (p 348). To find the volume of a parallelepiped in R^3 : (i) Find vectors u, v, w that determine the edges of the parallelepiped. (ii) Find the absolute value of the determinant of the 3 × 3 matrix with columns u, v, w. Note: By Ex 14 p 389 this is the absolute value of (u×v) · w, which computes the volume–see p 231.
USEFUL EXERCISES
(p 372 Ex #10). If k is a scalar and A is an n × n matrix, then det(kA) = kn^ det(A).
(p 372 Ex #19). Matrices in block form: If A and B are square matrices, then
A 0
C B
∣ =^ |A| · |B|.
(p 384 Ex #14). If A is an n × n matrix, then det(adj(A)) = [det(A)]n−^1.
Chapter 7: Eigenvalues and Eigenvectors
DEFINITIONS
7.1 (pp 394, 399). Let L : V → V be a linear transformation. The real number λ is an eigenvalue of L if there is a nonzero vector x in V for which L(x) = λx. In this case x is an eigenvector of L (with associated eigenvalue λ).
Note: this means that L maps the line determined by x back into itself.
Def (p 407, Ex 14). If λ is an eigenvalue of the linear transformation L : V → V , then the eigenspace associated with λ is {x in V | L(x) = λx}.
7.2 (p 401). The characteristic polynomial of a square matrix A is det(λI − A).
7.3 (p 410). A linear transformation L : V → V is diagonalizable if it is possible to find a basis S for V such that the matrix MS←S (L) of L with respect to S is a diagonal matrix.
Def (p 413). A square matrix is diagonalizable if it is similar to a diagonal matrix.
7.4 (p 430). A square matrix A is orthogonal if A−^1 = AT^.
THEOREMS
7.1 (p 402). The eigenvalues of a matrix are the real roots of its characteristic polynomial.
7.2 (p 411). Similar matrices have the same eigenvalues.
7.3 (p 412). A linear transformation L : V → V is diagonalizable if and only if it is possible to find a basis for V that consists of eigenvectors of L.
7.5 (p 415). A square matrix is diagonalizable if its characteristic polynomial has distinct real roots.
7.6 (p 427). The characteristic polynomial of a symmetric matrix has real roots.
7.7 (p 428). For a symmetric matrix, eigenvectors associated with distinct eigenvalues are orthogonal.
7.8 (p 431). A square matrix is orthogonal if and only if its columns (rows) are orthonormal.
7.9 (p 431). Let A be a symmetric matrix. Then there exists an orthogonal matrix P such that P −^1 AP is a diagonal matrix. The entries of P −^1 AP are the eigenvalues of A.
ALGORITHMS
- (p 406) To find the eigenvalues and eigenvectors of the matrix A: (i) Find the real solutions of the characteristic equation det(λI − a) = 0. (ii) For each value of λ from (i), solve the system (λI − A)x = 0.
- (p 436) To diagonalize the matrix A: (i) Find the eigenvalues of A by finding the real roots of its characteristic polynomial det(λI − A). (ii) For each eigenvalue λ, find a basis for its eigenspace by solving the equation (λI − A)x = 0. Note: The diagonalization process fails if the characteristic polynomial has a complex root or if it has a real root of multiplicity k whose eigenspace has dimension < k. (iii) P −^1 AP = D is a diagonal matrix if the columns of P are the basis vectors found in (ii) and D is the matrix whose diagonal entries are the eigenvalues of A (in exactly the same order).
- (p 436) To diagonalize the symmetric matrix A: Follow the steps in 2 , but use the Gram-Schmidt process to find an orthonormal basis for each eigenspace. Note: The diagonalization process always works, and P −^1 = P T^ since P is orthogonal.
