1. Let L, K : V → V be linear maps between finite-dimensional vector spaces that satisfy L o K = 0. Is it true that K o L = 0?

Q: (4) Suppose V and W are finite dimensional vector spaces such that dim(V) > dim(W). Show that any…

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Q: 1 Let S=(x | x=a +b +d 1 +c 1 1 where a,b,c,dER. Show explicitly that S is a vector space.

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Q: 4. Check that the following are vector spaces: (a) RN. 55 / 600 (b) CN. (c) C(N), where 2 is a…

A: A non-empty set V, +, · is said to be a vector space over a field F together with operations of…

Q: 1. Provide an example of a vector space Mmxn(Q) that is isomorphic to the vector space P19(Q).

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Q: Let S = {A2«2| trace(A) = 0}, i.e. S = Show that -a S is a vector space.

A: The solution is given as

Q: If T: V → W is a linear map between vector spaces V and W, T is one-to-one and onto, and a ba- sis…

A: Given that T:V→W is a linear map between the vector spaces V and W. Also, T is one-to-one and onto.…

Q: 6. Consider a normed vector space S = (R,|- |). Show that, for all r, y e S, |z - y| 2 | – l9|-

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Q: (3) The subset of pairs {(r, y) E R²|ry = 0} in the vector space R².

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Q: Suppose V is an inner-product space and u, v e V satisfy ||u|| = ||v|| = 1 and (u, v) = 1. Prove…

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Q: b) Let X be an inner product space and x, y e x.Prove that xly if and only if kx + y = |lk | + |y*k…

A: Given that X be an inner product space and x,y∈X Let, x⊥y. The objective is to prove that…

Q: Exercise 8.2. Let X, Y, Z be normed spaces and suppose that T E B(X, Y) and S e B(Y, Z). Show that…

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Q: 5) Let V, W be finite dimensional vector spaces over F. (a) ( Prove that (b) Use this to prove that…

A: Question is wrong • please see the justification

Q: et X be a normed linear space and T: X→ X and S: X→ X be two perators, then S T :X→X is a bounded…

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Q: If X is vector space and (X,d) is discrete metric space, then X can induce an inner product space…

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Q: Let E be a vector space. If E= U+V and E=U+W, then there is an isomorphism f: V--W between

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Q: 6) Prove that L(V,W) is a vector space under the usual addition and scalar multiplication of linear…

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Q: If a subset S of a vector space V is linearly independent, then S union any vector w, SU {w}, is…

A: Let 'V' be vector space over field 'F'  Standard basis  elements of ℝ3 are given as: (1,0,0),…

Q: Let S={x | x=a+b +c 1 where a,b,c,deR. Show explicitly that S is a vector space.

A: We will construct a transformation matrix whose span will be the set S, and the  we will show that…

Q: 10. Show that V = R² with the standard scalar multiplication, but addition defined by (x, , y.) +…

A: We have to show that the set V with the addition operation x1, y1+x2, y2=x13+x233, y13+y233 and…

Q: 1) The space (R²,O,O) is not a vector space where: (x, y) O (z, w) = (x + z + 1, y + w + 1) a O (x,…

A: A vector space is defined over field F that consists of a set on which two operations (addition and…

Q: Suppose that W is a finite-dimensional vector space over F and that T:V→W is linear. Prove that…

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Q: Show that V={(x,2020x): x in R} is a vector space

A: According to the given information, it is required to show that the given set is a vector space.

Q: 1. Suppose that x is a non-zero vector in a normed vector space V over F, and that a > 0. Find a…

A: Given : V is any normed vector space and x is any non zero vector and α>0. To find :  A real…

Q: 5. Show that a translation t by a vector (a, b) defined by t(x, y) = (x + a, y + b) is an isometry.

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Q: ) Let X be a finite dimensional norm space. Then prove that M = {x EX |||x || <1} is %3D compact.

A: Sufficient to prove that M is closed and bounded

Q: Suppose u and v belong to a vector space V. Simplify cach of the following cxpressions: (a) E = 4(5u…

A: Given u,v belongs to vector space W

Q: 1. Consider the real vector spaces C2 and R2×2, and let f: C2 → R²×² defined by f(x, y) = (4 'а — d…

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Q: If M = {x: Lx = 0}, where L is a continuous linear functional on H, prove that M¹ is a vector space…

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Q: If V is an n-dimensional vector space, W is an m-dimensional space, and T: V → W is an onto linear…

A: False.

Q: In any vector space V, show that (a + b)(x + y) = ax + ay + bx + by for any x, y e V and any a, be…

A: In any vector space V, show that (a+b)(x+y)=ax+ay+bx+by for any x,y belong to V and any a,b belong…

Q: السؤال 7 T(ax+By) = a Tx+B Ty ylgn O İhi O السؤال 8 The range R(T) is not vector space 0 صواب İhi O

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Q: Suppose that W1, W2,. , Wm and V are all finite dimensional vector spaces over F. Prove that 2,...…

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Q: Let V and W be vector spaces and T:V→W be linear. Let {y1,…,yk} be a linearly independent subset of…

A: It is given that Let V and W be vector spaces and T: V→W be linear. Let {y1… yk} be a linearly…

Q: If X is finite dimensional hor med space o T: X-Dy is a Linear operator from X to normed space y.…

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Q: Prove that the vector space C of all continuous functions from R to R is infinite-dimensional. Shew…

A: We have to prove given statement:

Q: 2. Prove or For finitely-generated vector spaces U, V, W disprove: r F, L(U, W) = L(V, W) if and…

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Q: Let V and W be finite-dimensional vector spaces and T: V →W be linear. (a) Prove that if dim(V)…

A: (a)Let B={v1,…,vn} be a basis for V. Then, C={T(v1),….,T(vn)} spans T(v)

Q: c) Prove that a subset W of R’ defined by W ={(x, y, z) e R² / 2x= y = 2z} is a vector space of R'.

A: Given, W = { (x,y,z) in R³ | 2x = y = 2z } is a subset of R³ Let, u = (x,y,z) &amp; v = (a,b,c)…

Q: The vector spaces M2, 2 and M5,1 are isomorphic to each other ylgn ihi

A: False

Q: (ii) Prove that W is a vector space of R°.

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Q: Q3) Let L = Z* and |x ||= max {]x1 - x3l, x2l, 15x,1} Vx = (x1, X2, X3, X4) E Z*. Is (Z*, ) is a…

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Q: 9) Suppose V is a vector space and S,T E L(V,V) are such that Range S c ker T =null T. Prove that…

A: See the detailed solution below.

Q: Suppose W1, ..., Wm are vector spaces. Prove that L(V, W1 × • × Wm) and L(V, W1) × ·…× L(V, Wm) are…

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Q: 12. Show that d((x1, Y1), (x2, Y2)) = |x1x2- Y1Y2| is not a metric on X = R %3D %3D

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Q: If V1, V2, V3, V4, and W are all vector spaces.  Are L(V1x...xV4, W) and L(V1, W) x...x L(V4, W)…

A: Given that: V1,V2,V3,V4,W=Vector space The objective is to verify the given statement. Using the…

Q: State if the following statement is true or false. If is a vector space and pM is the dual of v then…

A: If is V is a vector space with finite dimension then V is isomorphic to its dual D(V). But here…

Q: 14. If d is a metric on a vector space X‡ {0} which is obtained from a norm, and å is defined by…

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Q: Prove that the map || - || : L(V, W)→R defined by ||L|| = sup {|| L(x)|| | ||| < 1} is a norm on…

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Q: If S and T are two subsets of a vector space V (F) and S cL (T) then show that L(S) CL (T).

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Q: Consider E = C( [o,(]) as normed spaces with normS II f | = sup 1fcxl and l fll,=Sifwildx. %3D Show…

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Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!