Review of Linear Algebra - Carnegie Mellon University
... What is special about orthonormal bases? Projection is easy Very useful length property Universal (Gram Schmidt) given any basis can find an orthonormal basis that has the same span ...
... What is special about orthonormal bases? Projection is easy Very useful length property Universal (Gram Schmidt) given any basis can find an orthonormal basis that has the same span ...
18.03 LA.2: Matrix multiplication, rank, solving linear systems
... Many people think about taking the dot product of the rows. That is also a perfectly valid way to multiply. But this column picture is very nice because it gets right to the heart of the two fundamental operation that we can do with vectors. We can multiply them by scalar numbers, such as x1 and x2 ...
... Many people think about taking the dot product of the rows. That is also a perfectly valid way to multiply. But this column picture is very nice because it gets right to the heart of the two fundamental operation that we can do with vectors. We can multiply them by scalar numbers, such as x1 and x2 ...
Lecture 8: Examples of linear transformations
... In general, shears are transformation in the plane with the property that there is a vector w ~ such that T (w) ~ =w ~ and T (~x) − ~x is a multiple of w ~ for all ~x. Shear transformations are invertible, and are important in general because they are examples which can not be diagonalized. ...
... In general, shears are transformation in the plane with the property that there is a vector w ~ such that T (w) ~ =w ~ and T (~x) − ~x is a multiple of w ~ for all ~x. Shear transformations are invertible, and are important in general because they are examples which can not be diagonalized. ...
Linear algebra - Practice problems for midterm 2 1. Let T : P 2 → P3
... Solution: This is not a subspace because it does not contain the zero polynomial. (c) W = {p(x) : the coefficient of x2 in p(x) is 0}. Solution: This is a subspace, because if p(x), q(x) have no x2 term, then neither do p(x) + q(x) and rq(x) for r ∈ R. 3. Let Mm×n be the vector space of m × n matric ...
... Solution: This is not a subspace because it does not contain the zero polynomial. (c) W = {p(x) : the coefficient of x2 in p(x) is 0}. Solution: This is a subspace, because if p(x), q(x) have no x2 term, then neither do p(x) + q(x) and rq(x) for r ∈ R. 3. Let Mm×n be the vector space of m × n matric ...
4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular
... straight lines in place of brackets when writing matrices because the notation ...
... straight lines in place of brackets when writing matrices because the notation ...
Unitary Matrices and Hermitian Matrices
... Note that the vector (1 + 8i, 2 − 3i) was conjugated and transposed. Doing the matrix multiplication, (1 − 8i)a + (2 + 3i)b = 0. I can get a solution (a, b) by switching the numbers 1 − 8i and 2 + 3i and negating one of them: (a, b) = (2 + 3i, −1 + 8i). There are two points about the equation u · v ...
... Note that the vector (1 + 8i, 2 − 3i) was conjugated and transposed. Doing the matrix multiplication, (1 − 8i)a + (2 + 3i)b = 0. I can get a solution (a, b) by switching the numbers 1 − 8i and 2 + 3i and negating one of them: (a, b) = (2 + 3i, −1 + 8i). There are two points about the equation u · v ...
June 2014
... a) Define what it means for a matrix G to be a generator matrix, and a matrix H to be a parity check matrix, for a linear code C over a finite field F. b) Let G, H be full rank matrices over F, where G is k × n and H is (n − k) × n. Prove that there exists a code C with generator matrix G and parity ...
... a) Define what it means for a matrix G to be a generator matrix, and a matrix H to be a parity check matrix, for a linear code C over a finite field F. b) Let G, H be full rank matrices over F, where G is k × n and H is (n − k) × n. Prove that there exists a code C with generator matrix G and parity ...
Estimation of structured transition matrices in high dimensions
... Estimation in this setting requires some extra structure. The recent literature has given much attention to the case of sparse transition matrices and penalized estimation which adapts to the unknown sparsity pattern. In this talk, the transition matrices are well approximated by matrices with a com ...
... Estimation in this setting requires some extra structure. The recent literature has given much attention to the case of sparse transition matrices and penalized estimation which adapts to the unknown sparsity pattern. In this talk, the transition matrices are well approximated by matrices with a com ...
Exercise 4
... The standard method for solving for the vector x is Gaussian elimination. A particularly useful formulation in terms of numerical treatment is called LU decomposition. Here the matrix A is written as the product of a lower triangular matrix L and an upper triangular matrix U, A = L . U. Note that th ...
... The standard method for solving for the vector x is Gaussian elimination. A particularly useful formulation in terms of numerical treatment is called LU decomposition. Here the matrix A is written as the product of a lower triangular matrix L and an upper triangular matrix U, A = L . U. Note that th ...
