REVIEW FOR MIDTERM I: MAT 310 (1) Let V denote a vector space
... W is a linear transformation then T ( 3i=1 ai vi ) = 3i=1 ai T (vi ) is true for any real numbers ai and any vectors vi ∈ V . (c) If T : V −→ W is a linear transformation, then give the definition for N (T ) — the null space of T . Prove (from basics) that N(T) is a subspace of V. (6) Let T : R6 −→ ...
... W is a linear transformation then T ( 3i=1 ai vi ) = 3i=1 ai T (vi ) is true for any real numbers ai and any vectors vi ∈ V . (c) If T : V −→ W is a linear transformation, then give the definition for N (T ) — the null space of T . Prove (from basics) that N(T) is a subspace of V. (6) Let T : R6 −→ ...
Math39104-Notes - Department of Mathematics, CCNY
... case the determinant is zero, the vectors are linearly dependent and any 1 × 2 matrix W solves the system. Now we assume that A is not the zero matrix. If (a, b) = C(c, d) then det(A) = Ccd − Cdc = 0 and W = (1 − C) is a nonzero solution of the system. If W = (w1 w2 ) is a solution of the system the ...
... case the determinant is zero, the vectors are linearly dependent and any 1 × 2 matrix W solves the system. Now we assume that A is not the zero matrix. If (a, b) = C(c, d) then det(A) = Ccd − Cdc = 0 and W = (1 − C) is a nonzero solution of the system. If W = (w1 w2 ) is a solution of the system the ...
Reduced Row Echelon Form Consistent and Inconsistent Linear Systems Linear Combination Linear Independence
... 1. exactly one solution (no free variables), ...
... 1. exactly one solution (no free variables), ...
Parameter estimation in multivariate models Let X1,..., Xn be i.i.d.
... • If the covariance matrix of an unbiased estimator attains the Cramér–Rao information (matrix) limit (see the forthcoming definition), then it is the efficient estimator. • Even if the information limit cannot be attained with any unbiased estimator, there may exist an efficient estimator. As a co ...
... • If the covariance matrix of an unbiased estimator attains the Cramér–Rao information (matrix) limit (see the forthcoming definition), then it is the efficient estimator. • Even if the information limit cannot be attained with any unbiased estimator, there may exist an efficient estimator. As a co ...
Linear_Algebra.pdf
... 2. It is possible to multiply rectangular matrices as long as they conform and the result if another rectangular matrix. In the example above, matrices a and b do conform because a is a 2 2 and b is a 2 3: The product matrix c is a 2 3: Reversing the order of matrix multiplication in this case does ...
... 2. It is possible to multiply rectangular matrices as long as they conform and the result if another rectangular matrix. In the example above, matrices a and b do conform because a is a 2 2 and b is a 2 3: The product matrix c is a 2 3: Reversing the order of matrix multiplication in this case does ...
Pythagoreans quadruples on the future light cone
... Therefore it will be enough to prove that starting with any GPQ with d > 1, we can always reach one of the three elements of the generating set q1 , q2 , q3 . To prove that it will be enough to show that unless two elements are zero and the third is greater or equal to 1 (what implies d = 1), we can ...
... Therefore it will be enough to prove that starting with any GPQ with d > 1, we can always reach one of the three elements of the generating set q1 , q2 , q3 . To prove that it will be enough to show that unless two elements are zero and the third is greater or equal to 1 (what implies d = 1), we can ...
ULinear Algebra and Matrices
... of the equals sign and constants are on the right. Write the augmented matrix that corresponds to the system. Use row operations to transform the first column so that all elements except the element in the first row are zero. Use row operations to transform the second column so that all elements exc ...
... of the equals sign and constants are on the right. Write the augmented matrix that corresponds to the system. Use row operations to transform the first column so that all elements except the element in the first row are zero. Use row operations to transform the second column so that all elements exc ...
Approximating sparse binary matrices in the cut
... Proof of Theorem 1.1: Here is an outline of the proof. Let A be the n by n identity matrix, and n . We first show that B does not contain too suppose B is an n by n matrix so that ||A − B||C ≤ 16 many rows of Euclidean norm exceeding some absolute constant, and omit these rows, if any, and the corre ...
... Proof of Theorem 1.1: Here is an outline of the proof. Let A be the n by n identity matrix, and n . We first show that B does not contain too suppose B is an n by n matrix so that ||A − B||C ≤ 16 many rows of Euclidean norm exceeding some absolute constant, and omit these rows, if any, and the corre ...
In algebra, a determinant is a function depending on
... This means that the determinant is a similarity invariant. Because of this, the determinant of some linear transformation T : V → V for some finite dimensional vector space V is independent of the basis for V. The relationship is one-way, however: there exist matrices which have the same determinant ...
... This means that the determinant is a similarity invariant. Because of this, the determinant of some linear transformation T : V → V for some finite dimensional vector space V is independent of the basis for V. The relationship is one-way, however: there exist matrices which have the same determinant ...
Jordan normal form
In linear algebra, a Jordan normal form (often called Jordan canonical form)of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called a Jordan matrix, representing the operator with respect to some basis. Such matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal (on the superdiagonal), and with identical diagonal entries to the left and below them. If the vector space is over a field K, then a basis with respect to which the matrix has the required form exists if and only if all eigenvalues of the matrix lie in K, or equivalently if the characteristic polynomial of the operator splits into linear factors over K. This condition is always satisfied if K is the field of complex numbers. The diagonal entries of the normal form are the eigenvalues of the operator, with the number of times each one occurs being given by its algebraic multiplicity.If the operator is originally given by a square matrix M, then its Jordan normal form is also called the Jordan normal form of M. Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix. In spite of its name, the normal form for a given M is not entirely unique, as it is a block diagonal matrix formed of Jordan blocks, the order of which is not fixed; it is conventional to group blocks for the same eigenvalue together, but no ordering is imposed among the eigenvalues, nor among the blocks for a given eigenvalue, although the latter could for instance be ordered by weakly decreasing size.The Jordan–Chevalley decomposition is particularly simple with respect to a basis for which the operator takes its Jordan normal form. The diagonal form for diagonalizable matrices, for instance normal matrices, is a special case of the Jordan normal form.The Jordan normal form is named after Camille Jordan.