Suppose u, and v are vectors in R m, such that ∥u∥ = 1, ∥v∥ = 4, ∥u + v∥ = 5. Find the inner product 〈u, v〉.
Suppose {a1, · · · ak} are orthonormal vectors in R m. Show that {a1, · · · ak} is a linearly independent set.
In: Advanced Math
2) Let v, w, and x be vectors in Rn.
a) If v is the zero vector, what geometric object represents all
linear
combinations of v?
b) Same question as a), except now for a nonzero v.
c) Same question as a) except now for nonzero vectors v and w (be
care-
ful!).
d) Same question as a) except now for nonzero vectors v, w, and x
(be
extra careful!).
In: Advanced Math
Let V be the set of positive reals, V = {x ∈ R : x > 0}. Define “addition” on V by x“ + ”y = xy, and for α ∈ R, define “scalar multiplication” on V by “αx” = x^α . Is V a vector space with these unusual operations of addition and scalar multiplication? Prove your answer.
In: Advanced Math
Suppose T ∈ L(V)
a) Show that V = ImT0 ⊃ ImT1 ⊃ ImT2 ⊃ ...
b) Show that if m ≥ 0 is such that ImTm = ImTm + 1, then ImTm + k =ImTm + k + 1 for all k ≥ 0.
c) Show that if n = dim V, then ImTn = ImTn + 1 = ImTn + 2 = ....
In: Advanced Math
Let V and W be Banach spaces and suppose T : V → W is a linear map. Suppose that for every f ∈ W∗ the corresponding linear map f ◦ T on V is in V ∗ . Prove that T is bounded.
In: Advanced Math
The potential of a particle is given as V (x) = -Vδ (x), with V being a positive number. Find the wave function and energy of the bounded states of this particle.
In: Physics
Let v and w be two nonzero vectors in R4 . Then v and w are linearly independent if only if v is not a scalar multiple of w. True or false?
In: Advanced Math
Design an experiment to knock out a certain gene of interest in the human genome. Outline the experiment, explaining the methods you would use, and also indicate how you would verify your experiment was successful at each step.
In: Biology
Suppose in a certain cohort study a researcher followed 360 college students who regularly smoked cigarettes, and 1200 college students who did not regularly smoke cigarettes. None of the subjects had peptic ulcers at the start of the study period. At the end of the study period, 90 of the smokers and 60 of the nonsmokers had at least one peptic ulcer. For simplicity, assume that none of the students died or left the study early. Also, assume the sample is representative of the population of college students in Nevada. Finally, suppose that 20% of college students in Nevada regularly smoke cigarettes.
1. In the smokers, what percentage of the risk for developing at least one peptic ulcer is directly attributable to smoking?
2. In the population of Nevada college students, what absolute amount of risk for developing at least one peptic ulcer is directly attributable to smoking?
3. In the population of Nevada college students, what percentage decrease in the risk for developing at least one peptic ulcer could be achieved by completely eliminating smoking?
In: Statistics and Probability
Suppose in a certain cohort study a researcher followed 360 college students who regularly smoked cigarettes, and 1200 college students who did not regularly smoke cigarettes. None of the subjects had peptic ulcers at the start of the study period. At the end of the study period, 90 of the smokers and 60 of the nonsmokers had at least one peptic ulcer. For simplicity, assume that none of the students died or left the study early. Also, assume the sample is representative of the population of college students in Nevada. Finally, suppose that 20% of college students in Nevada regularly smoke cigarettes.
1. In the smokers, what percentage of the risk for developing at least one peptic ulcer is directly attributable to smoking?
2. In the population of Nevada college students, what absolute amount of risk for developing at least one peptic ulcer is directly attributable to smoking?
3. In the population of Nevada college students, what percentage decrease in the risk for developing at least one peptic ulcer could be achieved by completely eliminating smoking?
In: Statistics and Probability