D Linear Algebra: Determinants, Inverses, Rank
... is called a homogeneous system. From the rule (D.17) we see that if |A| is nonzero, all solution components are zero, and consequently the only possible solution is the trivial one x = 0. The case in which |A| vanishes is discussed in the next section. §D.3. Singular Matrices, Rank If the determinan ...
... is called a homogeneous system. From the rule (D.17) we see that if |A| is nonzero, all solution components are zero, and consequently the only possible solution is the trivial one x = 0. The case in which |A| vanishes is discussed in the next section. §D.3. Singular Matrices, Rank If the determinan ...
3.7.8 Solving Linear Systems
... number of independent equations less than the number of variables; no solutions or many solutions may exist number of equations more than the number of variables; solutions may or may not exist number of independent equations equal to the number of variables, and determinant non-zero; a unique solut ...
... number of independent equations less than the number of variables; no solutions or many solutions may exist number of equations more than the number of variables; solutions may or may not exist number of independent equations equal to the number of variables, and determinant non-zero; a unique solut ...
6.4 Krylov Subspaces and Conjugate Gradients
... The extreme eigenvalues of this Hilbert matrix are �max � 1.5 and �min � 10−4 . As always, those are the squares of the singular values αmax and αmin of Vc . The condition number of the power basis 1, x, x2 , x3 is the ratio αmax /αmin � 125. If you want a more impressive number (a numerical disaste ...
... The extreme eigenvalues of this Hilbert matrix are �max � 1.5 and �min � 10−4 . As always, those are the squares of the singular values αmax and αmin of Vc . The condition number of the power basis 1, x, x2 , x3 is the ratio αmax /αmin � 125. If you want a more impressive number (a numerical disaste ...
n-Dimensional Euclidean Space and Matrices
... Definition of n space. As was learned in Math 1b, a point in Euclidean three space can be thought of in any of three ways: (i) as the set of triples (x, y, z) where x, y, and z are real numbers; (ii) as the set of points in space; (iii) as the set of directed line segments in space, based at the ori ...
... Definition of n space. As was learned in Math 1b, a point in Euclidean three space can be thought of in any of three ways: (i) as the set of triples (x, y, z) where x, y, and z are real numbers; (ii) as the set of points in space; (iii) as the set of directed line segments in space, based at the ori ...
Exercise Set iv 1. Let W1 be a set of all vectors (a, b, c, d) in R4 such
... 1. Let W1 be a set of all vectors (a, b, c, d) in R4 such that a + d = 0 and W2 be a set of all vectors (a, b, c, d) in R4 such that ad = 0. Is W1 a subspace of R4 ? Is W2 a subspace of R4 ? 2. Let u~1 = (1, 1, 0), u~2 = (1, 0, 1), u~3 = (0, 1, 1) (a) Show that {u~1 , u~2 , u~3 } is linearly indepen ...
... 1. Let W1 be a set of all vectors (a, b, c, d) in R4 such that a + d = 0 and W2 be a set of all vectors (a, b, c, d) in R4 such that ad = 0. Is W1 a subspace of R4 ? Is W2 a subspace of R4 ? 2. Let u~1 = (1, 1, 0), u~2 = (1, 0, 1), u~3 = (0, 1, 1) (a) Show that {u~1 , u~2 , u~3 } is linearly indepen ...
1 The Covariance Matrix
... subspace and we can select orthogonal vectors spanning this subspace. So there always exists an orhtonormal set of eigenvectors of Σ. It is often convenient to work in an orthonormal coordinate system where the coordinate axes are eigenvectors of Σ. In this coordinte system we have that Σ is a diag ...
... subspace and we can select orthogonal vectors spanning this subspace. So there always exists an orhtonormal set of eigenvectors of Σ. It is often convenient to work in an orthonormal coordinate system where the coordinate axes are eigenvectors of Σ. In this coordinte system we have that Σ is a diag ...
Solutions to Math 51 First Exam — January 29, 2015
... MAYBE (e) Given a matrix A, the column space of A is equal to the column space of T F the reduced row echelon form of A. On the one hand, consider any A already in RREF, for which certainly C(A) = C(rref(A)); on the other, consider A = [ 11 11 ], for which C(A) = span ([ 11 ]) but C(rref(A)) = C ([ ...
... MAYBE (e) Given a matrix A, the column space of A is equal to the column space of T F the reduced row echelon form of A. On the one hand, consider any A already in RREF, for which certainly C(A) = C(rref(A)); on the other, consider A = [ 11 11 ], for which C(A) = span ([ 11 ]) but C(rref(A)) = C ([ ...
