Heating Water : Modeling with a Linear Equation

Water is the most importnt substance on Earth. One reason for its usefulness is that is exists as a liquid over a wide range of temp. In its liquid range, water absorbs or releases heat directly in proportion to its change in temp. Consider the following data that shows temp of a 1,000 g sampe of water at normal atmospheric pres ...continues

Systems of Equations Word Problems

Supppose a baseball is thrown at 85 miles per hour.The ball will travel 320 ft when hit by a bat swung at 50 miles per hour and will travel 440 ft when hit by a bat swung at 80 miles per hour. Let y be the number of ft traveled by the ball when hit by a bat swung at x miles per hour.(Note: The precceding data is valid for 50 les ...continues

Consider the following data that show temperature

Consider the following data that show temperature of a 1,000 g sample of water at normal atmospheric pressure as a function of heat supplied. A kJ can simply be thought of a unit of heat. Temperature Heat Supplied 0 oC 0 kJ 10 oC 42 kJ 30 oC 126 kJ 50 oC 209 kJ 80 oC 335 kJ 99 oC 414 kJ 100 0C 420 kJ Base ...continues

Intepreting Data and Piecewise Functions

A factory begins emitting particulate matter into the atmosphere at 8 am each workday, with the emissions continuing until 4 pm. The level of pollutants, P(t), measured by a monitoring station 1/2 mile away is approximated as follows, where t represents the number of hours since 8 am: p(t)= 75t + 100 if 0 is les ...continues

Linear Operators : Finite-dimensional Vector Space, Fields and Mappings

Let V be a finite-dimensional vector space. The base field F may be either R or C here. Let T, an element of the linear mapping of V to V, L(V), be an operator. Suppose that all non-zero elements of V are eigenvectors for T. Show that T is a scalar multiple of the identity map, i.e., that there is a λ in the Reals such ...continues

Let n be a positive integer. Let A be an element of the vector space Mat(n,n,F), which has dimension n2 over F. Show that the span of the infinite set of matrices span(In, A, A2, A3, …) has dimensio

Let n be a positive integer. Let A be an element of the vector space Mat(n,n,F), which has dimension n2 over F. Show that the span of the infinite set of matrices span(In, A, A2, A3, …) has dimension not exceeding n over F. Defn of the linear space Mat(n,n,F): The set of all n-by-n matrices with entries in F. Mat(n,n,F ) is ...continues

Complex Inner-Product Space : Complex Spectral Theorem

Suppose that V is a complex (i.e. F = C) inner-product space. Prove that if N, an element of L(V), is normal and nilpotent, then N = 0. Use Complex Spectral Theorem: Suppose that V is a complex inner-product space and T is an element of L(v). Then V has an orthonormal basis consisting of eigenvectors of T if and only if T ...continues

Standard Inner Product and Linear Operators

This is for #7 section 8.1 in Hoffman and Kunze's Linear Algebra book. Any help and detailed explanations will be greatly appreciated! Thanks! Please see attached file for full problem description.

Vector Subspace, Orthonormal Basis, Othogonal Projection and Inner Product

This is for #6 section 8.2 in Hoffman and Kunze's Linear Algebra book. Any help and detailed explanations will be greatly appreciated! Thanks. Please see attached file for full problem description.

Inner Product Space, Linear Operators, Commutativity and Standard Inner Product

This is for #2 & 4 section 8.3 in Hoffman and Kunze's Linear Algebra book. Any help and detailed explanations will be greatly appreciated! Please see the attached file for the fully formatted problems.