Matrix vector product - Samenvattingen en Aantekeningen
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This is the perfect document for you to practice. It contains all the solutions that you need to practice for the end-term.
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Exam (elaborations) TEST BANK FOR The Chemistry Maths Book 2nd Edition
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Exam (elaborations) TEST BANK FOR The Chemistry Maths Book 2nd Edition 
Numbers, variables, and units 1 
1.1 Concepts 1 
1.2 Real numbers 3 
1.3 Factorization, factors, and factorials 7 
1.4 Decimal representation of numbers 9 
1.5 Variables 13 
1.6 The algebra of real numbers 14 
1.7 Complex numbers 19 
1.8 Units 19 
1.9 Exercises 29 
2 Algebraic functions 31 
2.1 Concepts 31 
2.2 Graphical representation of functions 32 
2.3 Factorization and simplification of expressions 34 
2.4 Inverse func...
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Exam (elaborations) TEST BANK FOR Manifolds, Tensor and Forms An Intro
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Exam (elaborations) TEST BANK FOR Manifolds, Tensor and Forms An Intro 
6 Let v1, v2 ∈ ker T . Then T (av1 + bv2) = aT v1 + bT v2 = 0, so ker T is 
closed under linear combinations. Moreover ker T contains the zero vector of 
V. All the other vector space properties are easily seen to follow, so ker T is a 
subspace of V. Similarly, let w1, w2 ∈ im T and consider aw1 + bw2. There 
exist v1, v2 ∈ V such that T v1 = w1 and T v2 = w2, so T (av1 + bv2) = 
aT v1 + bT v2 = aw1 + bw2, which show...
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Exam (elaborations) TEST BANK FOR Microeconomic Analysis 3rd Edition B
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Exam (elaborations) TEST BANK FOR Microeconomic Analysis 3rd Edition B 
1 False. There are many counterexamples. Consider the technology 
generated by a production function f(x) = x2. The production set is 
Y = {(y, −x) : y ≤ x2} which is certainly not convex, but the input requirement set is V (y) = {x : x ≥ √y} which is a convex set. 
1.2 It doesn’t change. 
1.3 1 = a and 2 = b. 
1.4 Let y(t) = f(tx). Then 
dy 
dt = Xn 
i=1 
∂f(x) 
∂xi 
xi, 
so that 
1 
y 
dy 
dt = 1 
f(x) 
X...
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Exam (elaborations) TEST BANK FOR Introduction to Solid State Physics
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Exam (elaborations) TEST BANK FOR Introduction to Solid State Physics 
The vectors xˆ ˆ + + y zˆ and − −xˆ yˆ + zˆ are in the directions of two body diagonals of a 
cube. If θ is the angle between them, their scalar product gives cos θ = –1/3, whence 
. 1 cos 1/ 3 90 19 28' 109 28' − θ = = ° + ° = ° 
2. The plane (100) is normal to the x axis. It intercepts the a' axis at and the c' axis 
at ; therefore the indices referred to the primitive axes are (101). Similarly,...
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Exam (elaborations) TEST BANK FOR Introduction to Solid State Physics
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Exam (elaborations) TEST BANK FOR Introduction to Solid State Physics 
The vectors xˆ ˆ + + y zˆ and − −xˆ yˆ + zˆ are in the directions of two body diagonals of a 
cube. If θ is the angle between them, their scalar product gives cos θ = –1/3, whence 
. 1 cos 1/ 3 90 19 28' 109 28' − θ = = ° + ° = ° 
2. The plane (100) is normal to the x axis. It intercepts the a' axis at and the c' axis 
at ; therefore the indices referred to the primitive axes are (101). Similarly,...
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