Math
Introduction to Linear Algebra 4th Edition solutions
Author: Gilbert Strang
Publisher: Wellesley Cambridge Press
ISBN: 9780980232714
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Key Math Terms and definitions covered in this textbook
Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).
Affine transformation
Tv = Av + Vo = linear transformation plus shift.
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
Cayley-Hamilton Theorem.
peA) = det(A - AI) has peA) = zero matrix.
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.
Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
Cofactor Cij.
Remove row i and column j; multiply the determinant by (-I)i + j •
Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).
Column space C (A) =
space of all combinations of the columns of A.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
Companion matrix.
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
Complex conjugate
z = a - ib for any complex number z = a + ib. Then zz = Iz12.