Partial Solutions for Linear Algebra by Friedberg et al . Chapter 1
نویسنده
چکیده
1.1.8. In any vector space V , show that (a+ b)(x+y) = ax+ay+ bx+ by for any x, y ∈ V and any a, b ∈ F . Proof. Let x, y ∈ V and a, b ∈ F . Note that (a + b) ∈ F (it is a scalar). By (VS7 ), we have that (a + b)(x + y) = (a + b)x + (a + b)y. By (VS8 ), (a + b)x = ax + bx. Likewise, (a + b)y = ay + by. Thus, (a+ b)x+ (a+ b)y = ax+ bx+ ay + by. Finally, by (VS1 ), we have (a+ b)(x+ y) = ax+ ay + bx+ by.
منابع مشابه
Application of the new extended (G'/G) -expansion method to find exact solutions for nonlinear partial differential equation
In recent years, numerous approaches have been utilized for finding the exact solutions to nonlinear partial differential equations. One such method is known as the new extended (G'/G)-expansion method and was proposed by Roshid et al. In this paper, we apply this method and achieve exact solutions to nonlinear partial differential equations (NLPDEs), namely the Benjamin-Ono equation. It is est...
متن کاملOn some open problems in cone metric space over Banach algebra
In this paper we prove an analogue of Banach and Kannan fixed point theorems by generalizing the Lipschitz constat $k$, in generalized Lipschitz mapping on cone metric space over Banach algebra, which are answers for the open problems proposed by Sastry et al, [K. P. R. Sastry, G. A. Naidu, T. Bakeshie, Fixed point theorems in cone metric spaces with Banach algebra cones, Int. J. of Math. Sci. ...
متن کاملAn L1-norm method for generating all of efficient solutions of multi-objective integer linear programming problem
This paper extends the proposed method by Jahanshahloo et al. (2004) (a method for generating all the efficient solutions of a 0–1 multi-objective linear programming problem, Asia-Pacific Journal of Operational Research). This paper considers the recession direction for a multi-objective integer linear programming (MOILP) problem and presents necessary and sufficient conditions to have unbounde...
متن کاملTorus Doubling Cascade in Problems with Symmetries
In this paper a multiple Hopf bifurcation is considered on the branch of trivial solutions in problems with O(2) symmetry. The theory for this type of bifurcation is well understood and leads us to solutions in the form of standing and travelling waves (van Gils and Mallet-Paret [1], Golubitsky et al [2]). Golubitsky et al [2] distinguish the standing and the travelling (rotating) oscillations ...
متن کاملEffect of Different Nutrient Solutions on The Yield, Chemical Composition and Nitrate Accumulation of Lettuce in Soilless Culture System
An experiment was conducted to select the optimum nutrient solution for growing lettuce (Lactuca sativa L.) under hydroponic conditions (open system) and Shahrekord climate, as a completely randomized design with five different nutrient solutions and three replicates, in Shahrekord University. The nutrient solutions were: 1) Research Station for Floriculture and Greenhouse Vegetables (the Nethe...
متن کامل