Linear Algebra A Summary

Transcription

1 Linear Algebra A Summary Definition: A real vector space is a set V that is provided with an addition and a multiplication such that (a) u V and v V u + v V, (1) u + v = v + u for all u V en v V, (2) u + (v + w) = (u + v) + w for all u V, v V, and w V, (3) there exists 0 V such that u + 0 = u for all u V, (4) for all u V there exists u V such that u + ( u) = 0, (b) u V and c R cu V, (5) c(u + v) = cu + cv for all u V, v V, and c R, (6) (c + d)u = cu + du for all u V, c R en d R, (7) c(du) = (cd)u for all u V, c R, and d R, (8) 1u = u for all u V. Definition: Let V be a vector space and W V. W is a subspace of V if W is a vector space with respect to the operations in V. Theorem: Let V be a vector space and W V, where W is not empty. If (a) u W and v W u + v W, (b) u W and c R cu W, then W is a subspace of V. Definition: Let V be a vector space and v 1, v 2,..., v n V. A vector v is a linear combination of v 1, v 2,..., v n if v = a 1 v 1 + a 2 v a n v n for some a 1, a 2,..., a n R. Definition: Let V be a vector space, v 1, v 2,..., v n V and S = {v 1, v 2,..., v n }, then span S is the span of S, i.e. span S = {a 1 v 1 + a 2 v a n v n a 1, a 2,..., a n R}. 1

2 Definition: Let V be a vector space and v 1, v 2,..., v n V. The vectors v 1, v 2,..., v n are linearly independent if a 1 v 1 + a 2 v a n v n = 0 a 1 = a 2 =... = a n = 0. Theorem: Let V be a vector space and S and T be finite subsets of V, where S T, then T is linearly independent S is linearly independent. Definition: Let V be a vector space. A basis of V is a set {v 1, v 2,..., v n }, where v 1, v 2,..., v n V, such that (a) V = span{v 1, v 2,..., v n }, (b) v 1, v 2,..., v n are linearly independent. Theorem: Let V be a vector space and S be a basis of V, then each vector v in V can be written as a unique linear combination of vectors in S. Note: We only consider vector spaces V that have a basis, or V = {0}. Theorem: Let V be a vector space and S be a finite subset of V, where span S = V, then some subset of S is a basis of V. Theorem: Let V be a vector space. If {v 1, v 2,..., v n } and {w 1, w 2,..., w m } are bases of V, then n = m. Definition: Let V be a vector space, where V {0}, then the dimension of V with notation dim V is the number of vectors of a basis of V. We define dim {0} = 0. Definition: Let V be a vector space and S = {v 1, v 2,..., v n } be an ordered basis of V. Let v V, then a 1 a 2 [v] S =., a n where v = a 1 v 1 + a 2 v a n v n, is the coordinate vector of v with respect to the ordered basis S. The elements of [v] S are the coordinates of v with respect to the ordered basis S. Definition: An m n-matrix A is a rectangular array of real numbers arranged in m rows and n columns, i.e. a 11 a a 1n a 21 a a 2n A =.... a m1 a m2... a mn 2

3 For j = 1, 2,..., n the j-th column of A is given by a 1j a 2j a j =.. a mj Theorem: The set of m n-matrices that is provided with an additive and a multiplicative operation a 11 a a 1n b 11 b b 1n a 21 a a 2n... + b 21 b b 2n... a m1 a m2... a mn b m1 b m2... b mn = c a 11 + b 11 a 12 + b a 1n + b 1n a 21 + b 21 a 22 + b a 2n + b 2n... a m1 + b m1 a m2 + b m2... a mn + b mn a 11 a a 1n a 21 a a 2n... = a m1 a m2... a mn, ca 11 ca ca 1n ca 21 ca ca 2n... ca m1 ca m2... ca mn is a vector space. The notation of this vector space is R m n. Also we denote R m = R m 1. Definition: Let A = (a ij ) be an m k-matrix and B = (b ij ) be a k n-matrix, then the matrix product AB is the m n-matrix C = (c ij ) with the entries c ij = a i1 b 1j + a i2 b 2j + + a ik b kj for i = 1, 2,..., m and j = 1, 2,..., n., Definition: Let A = ( a 1 a 2... ) a n be an m n-matrix, then the range of A is given by range A = span {a 1, a 2,..., a n }, and the rank of A is given by rank A = dim range A. Definition: Let A be an m n-matrix, then the kernel of A is given by ker A = {x R n Ax = 0}, and the nullity of A given by null A = dim ker A. 3

4 Theorem: Let A be an m n-matrix, then rank A + null A = n. Definition: An n n-matrix A is invertible (non-singular) if Ax = 0 x = 0 for all x R n. The inverse matrix A 1 is given by x = A 1 y y = Ax. Definition: Let V be a vector space with ordered bases S and T, then the transition matrix P S T from T to S is given by [v] S = P S T [v] T for all v V. Theorem: Let V be a vector space with ordered bases S = {v 1, v 2,..., v n } and T = {w 1, w 2,..., w n }, then P S T = ( [w 1 ] S [w 2 ] S... [w n ] S ). The matrix P S T is invertible, where P 1 S T = P T S. Definition: Let V be a vector space. A function : V R is a norm on V if (a) u 0 for all u V ; u = 0 u = 0, (b) u + v u + v for all u V and v V, (c) cu = c u for all u V and c R. A normed vector space is a vector space provided with a norm. Definition: Let V be a vector space. A function (, ) : V V R is an inner product on V if (a) (u, u) 0 for all u V ; (u, u) = 0 u = 0, (b) (v, u) = (u, v) for all u V and v V, (c) (u + v, w) = (u, w) + (v, w) for all u V, v V, and w V, (d) (cu, v) = c(u, v) for all u V, v V, and c R. 4

5 An inner product space is a vector space provided with an inner product. Theorem: Let V be an inner product space, then the Cauchy-Schwarz inequality holds: (u, v) (u, u)(v, v) for all u V en v V. Theorem: Let V be an inner product space and u = (u, u) for all u V, then V is a normed vector space. Definition: Let V be an inner product space with an ordered basis S = {v 1, v 2,..., v n }, then the inner product matrix A = (a ij ) with respect to S is given by a ij = (v j, v i ) for i, j = 1, 2,..., n. Definition: Let A = (a ij ) be an m n-matrix, then the transposed matrix is the n m-matrix A T = (a ji ). Definition: A square matrix A is symmetric if A T = A. Theorem: Let V be an inner product space with an ordered basis S and A the inner product matrix with respect to S, then (a) A is symmetric, (b) (v, w) = [v] T S A[w] S for all v V and w V. Definition: Let V be an inner product space, then the vectors u and v in V are orthogonal if (u, v) = 0. Definition: Let V be an inner product space with a basis S = {v 1, v 2,..., v n }, then S is orthonormal if (u j, u i ) = δ ij for i, j = 1, 2,..., n. Theorem: Let V be an inner product space with an ordered basis S, then (u, v) = [u] T S [v] S for all u V en v V. Definition: Let V be an inner product space with a basis {u 1, u 2,..., u n }, then the modified Gram-Schmidt process is given by v 1 = u 1 / u 1 and v k = u k (u k, v 1 )v 1 (u k, v 2 )v 2... (u k, v k 1 )v k 1 u k (u k, v 1 )v 1 (u k, v 2 )v 2... (u k, v k 1 )v k 1 5

6 for k = 2,..., n. Theorem: Let V be an inner product space with a basis {u 1, u 2,..., u n }, then the modified Gram-Schmidt process results into an orthonormal basis {v 1, v 2,..., v n }. Definition: Let V be an inner product space and W a subspace of V. complement W of W is given by The orthogonal u W (u, v) = 0 for all v W. Definition: Let V be a vector space and W 1 and W 2 subspaces of V, where W 1 W 2 = {0}, then the direct sum of W 1 and W 2 is given by W 1 W 2 = {w 1 + w 2 w 1 W 1 and w 2 W 2 }. Theorem: Let V be an inner product space and W a subspace of V, then V = W W. Theorem: Let A be an m n-matrix, then (a) ker A T = (range A), (b) range A T = (ker A). Definition: Let A R m n and b R m, then Ax = b with unknown vector x R n is a system of linear equations. This system is consistent if Ax = b for some vector x R n. The solution of the system is the set {x R n Ax = b}. Theorem: Let A R m n and b R m, then Ax = b is consistent b range A. Definition: A matrix is in the reduced row echelon form if: (a) There are only zero rows at the bottom of the matrix. (b) The first nonzero entry of a nonzero row is called the pivot of the row. (c) All entries left and under a pivot are equal to 0. Definition: An elementary row operation of a matrix is one of the following operations: (a) Interchange two rows. 6

7 (b) Multiply a row with a nonzero number. (c) Add a multiple of a row to another row. Definition: A matrix A is row equivalent with a matrix B if B can be obtained from A by elementary row operations. Theorem: A m n-matrix A is row equivalent to a matrix B if B = P A for some invertible m m-matrix P. Theorem: If the matrices A and B are row equivalent, then (a) ker A = ker B, (b) rank A = rank B. Theorem: Let A be a matrix in reduced row echelon form, then the norm of A is equal to the number of pivots. Definition: Let A R m n and b R m, then the system Ax = b is row equivalent to the system Bx = c if the matrix ( B c ) is row equivalent to ( A x ). Theorem: Row equivalent systems of linear equations have the same solution. Definition: Let S = [ n ], then a permutation of S is a rearrangement of the elements of S. Theorem: Let S = [ n ], then each permutation of S can be obtained from S by successive interchanges of elements. Definition: Let S = [ n ] and a permutation of S is obtained by n successive interchanges of rows, then the permutation is even or odd respectively if n is even or odd. Definition: Let A = (a ij ) be an n n-matrix, then the determinant of A is given by det A = (±)a 1j1 a 2j2... a njn, where the summation is over all permutations [ j 1 j 2... j n ] of the set [ n ]. The sign is + or if the permutation [ j1 j 2... j n ] respectively is even or odd. Theorem: Let A be an n n-matrix, then det A T = det A. Definition: An n n-matrix A = (a ij ) is an upper triangular matrix if i > j a ij = 0. 7

8 Theorem: Let the n n-matrix A = (a ij ) be an upper triangular matrix, then det A = a 11 a a nn. Theorem: Let the n n-matrix A be row equivalent with a matrix B, where B can be obtained from A by elementary row operations without row multiplications and with k row interchanges, then det A = ( 1) k det B. Theorem: Let A be an n n-matrix, then A is singular det A = 0. Definition: An n n-matrix A = (a ij ) is a diagonal matrix if i j a ij = 0. Theorem: Each invertible matrix is row equivalent to a diagonal matrix with nonzero diagonal entries. Theorem Let A and B be n n-matrices, then det (AB) = det A det B. Theorem: Let A be an invertible matrix, dan det A 1 = 1 det A. Definition: The Euclidean norm on R n is given by x 2 = x T x for all x R n. Definition: Let A R m n and b R m, then the vector x is a least squares solution of the system Ax = b if b A x 2 b Ax 2 for all x R n. Theorem: Let A R m n and b R m, then x is a least squares solution of Ax = b A T A x = A T b. Theorem: Let A R m n and b R m. If rank A = n, then A T A is invertible and Ax = b has a unique least squares solution x = (A T A) 1 A T b. 8

9 Definition: Let A be an n n-matrix, then the number λ is an eigenvalue of A corresponding to an eigenvector x, where x 0 if Ax = λx. Definition: An identity matrix is an n n-matrix I = (δ ij ). Definition: Let A be an n n-matrix, then the characteristic polynomial of A is given by p(λ) = det (λi A) for all λ R. Theorem: Let A be an n n-matrix, then the eigenvalues of A are the roots of the characteristic polynomial of A. Definition: Let A and B be n n-matrices, then B is similar to A if B = P 1 AP for some invertible n n-matrix P. Theorem: Similar matrices have equal eigenvalues. Definition: An n n-matrix A is diagonalizable if A is similar to a diagonal matrix. Theorem: Let A be an n n-matrix, then A is diagonalizable A has n linearly independent eigenvectors. Theorem: If the characteristic polynomial of an n n-matrix n has different roots, then A i diagonalizable. Theorem: A symmetric n n-matrix has n orthogonal eigenvectors. Definition: An n n-matrix is orthogonal if A T A = I. Theorem: Let A be a symmetric matrix, then there exists a diagonal matrix D and an orthogonal matrix P such that AP = P D. 9