1 Linear Transformations
... 1. Theorem 11: Suppose T : Rn → Rm is a linear transformation. Then T is one-to-one if and only if the equation T (x) = 0 has only the trivial solution. 2. Proof: First suppose that T is one-to-one. Then the transformation T maps at most one input vector in Rn to the output vector 0. Thus the equati ...
... 1. Theorem 11: Suppose T : Rn → Rm is a linear transformation. Then T is one-to-one if and only if the equation T (x) = 0 has only the trivial solution. 2. Proof: First suppose that T is one-to-one. Then the transformation T maps at most one input vector in Rn to the output vector 0. Thus the equati ...
Complex inner products
... Hermitian. The spectral theorem applies to Hermitian matrices and indeed it is most easily proven for Hermitian matrices. Since Lay does not provide a proof of the spectral theorem I will sketch a proof below. Theorem 1. If T : V → V is a linear transformation of a nonzero finite dimensional complex ...
... Hermitian. The spectral theorem applies to Hermitian matrices and indeed it is most easily proven for Hermitian matrices. Since Lay does not provide a proof of the spectral theorem I will sketch a proof below. Theorem 1. If T : V → V is a linear transformation of a nonzero finite dimensional complex ...
Solutions to Homework 2
... 2. Prove that the set of functions β = {1, cos x, cos2 x, . . . , cos6 x} is linearly independent in C(R). (Hint: Suppose c0 · 1 + c1 · cos x + · · · + c6 · cos6 x = 0. Then this equation holds for all values of x. Hence, for each value you substitute in for x, you get a different linear equation in ...
... 2. Prove that the set of functions β = {1, cos x, cos2 x, . . . , cos6 x} is linearly independent in C(R). (Hint: Suppose c0 · 1 + c1 · cos x + · · · + c6 · cos6 x = 0. Then this equation holds for all values of x. Hence, for each value you substitute in for x, you get a different linear equation in ...
Coding Theory: Homework 1
... Suppose we have an [n, k, d]2m with full rank G generating matrix of size k × n, which exists by Theorem 2.2.1. Create a new matrix G0 of size km × nm by the following procedure: For each a(i, j) in G will be replaced by an m × m block matrix. The first row will be the representation of a(i, j) unde ...
... Suppose we have an [n, k, d]2m with full rank G generating matrix of size k × n, which exists by Theorem 2.2.1. Create a new matrix G0 of size km × nm by the following procedure: For each a(i, j) in G will be replaced by an m × m block matrix. The first row will be the representation of a(i, j) unde ...
