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

Q: The set of vectors in R? with y = -3x is a real vector space

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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: 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: Suppose that the set A ={X₁, X2..., Xn}, X₁ ER" spans the entire space R", i.e. for any arbitrary n…

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Q: 5. Find a vector æ € R³ such that ||Ax – || is minimal, where [1 1 -2] 1 and b:= 2 1 1

A: By least square approximation ,      To minimize Ax-b the vector x is given by,…

Q: Suppose V₁, Vm, and W are vector spaces. Prove that L(V₁ × ... X Vm, W) are isomorphic to L(V₁, W) ×…

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Q: Suppose Y is a closed subspace of a normed linear space X and xo ¤ Y, x, € X . Prove that there is a…

A: Y is a closed subspace of a normed linear space X and x0∉Y, x0∈X.. We have to prove that there is a…

Q: Let V and W be vector spaces, let T : V --> W be linear, and let { w1 , w2 , ... , wk} be a…

A: Given:          V and W are vector spaces and T : V → W be linear.          The set w1, w2, ..., wk…

Q: Q4) Let C(R) be a normed space such that || f -sup{ Ifl,x E R}and T: C (R) -R define Tf)-If I. Is T…

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Q: Linear algebra

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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: 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: a) T: M, → R, where M, is the vector space of n by n matrices over a field, and T (A) = det(A).

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Q: Q2 In each of the following evaluate ||x + y|| of the given vectors in the given normed space and…

A: As per our guidelines we are supposed to answer only one question. Kindly repost other question as…

Q: Consider E =IR as a vector space. Show that: A function f: E → IR is a norm if and only if fu) =klxı…

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Q: Let V be a vector space, and let T : V → M₂2 be linear. Let V₁, V₂ EV and suppose 2 T(v₁) = J 1…

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Q: Find the boundary of the sequence xn in the normalized vector space C [a, b], if it exists. 4 cos 3…

A: The given sequence is: xnt=4cosn3t5+cosnt a=-5π, b=-4π

Q: 4. Consider the real vector space R?. For every x = (11,12) E R?, let ||x|| = |11|+ |r2l. a) Show…

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Q: (6) Suppose that T is a linear operator on a 2 dimensional vector space V and that T+ a I for any a…

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Q: Suppose V is a vector space and S, T = L(V, V) are such that Range Scker T=null T. Prove that (ST)²…

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Q: (b) If X is a normed line- ball M= {xeX:x≤1}

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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: Let X and Y be normed linear spaces, and T E B(X,Y). Prove that 38 > 0 such that 8||x|| < ||T(æ)||,…

A: Consider the given information.

Q: (10) Let the vector space P, has the inner product (p(x), q(x)) = | P(x)q(x)dx (a) Compute ||p(x)||…

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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 that v is a vector in R" such that u - v = 0 for every u e R". Prove that v = 0.

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Q: Given that U is a function of x, y and z and A a vector field, prove that: V· (UA) = (VU) · A + U(V…

A: We have to prove the given property.

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: 32. In a normed vector space (V, ||) show that B,(x) = x + B,(0) = {x + y: Ilyll <r} and that B,(0)…

A: Definition : Let ( V , || . || ) be a normed vector space .For any r &gt; 0 ,  the set Br (x)={ y ∈V…

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: Theorem 2: Let X be a finite dimensional normed space and let r > 0. Then, the closed unit ball B[0;…

A: Theorems and definitions we are going to use: 1. Definition: (X,|| . ||) is said to be bounded if…

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: Let X, Y be normed linear spaces. On X x Y define || (x,y) |l, = || × |lx + || y lly and + || y ily…

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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: Suppose that the set A={X₁, X2..., Xn}, X₁ ER" spans the entire space R", i.e. for any arbitrary…

A: Given: The set A=x1,x2,...,xn, xi∈ℝn spans the entire space ℝn, i.e. for any arbitrary n-vector…

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: A Consider E = \R as a vector space. Show that: if and only if fixed real winber A function f: E →…

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Q: Therefore (0,0, 0) e W. Ex 6. If W, and W2 be any two sub-spaces of a vector space V, then inW, is…

A: We have to prove that intersection of two subspace of V  is also a subspace of V

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 the vectors x"(t) =( t and x (t) = ( 4t 6. (a) Compute the Wronskian of x) and x2). (1) W =…

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Q: 3 Let W = span Given y = 1 decompose this vector into y in W and y, in w. 3 13-

A: See the detailed solution below.

Q: Let X is a normed linear space and the ball M= {xe X:|| x ||≤1} is compact, then X is finite…

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Q: 10. Suppose that V is a normed vector space over F. Let S = {x € V : ||x|| 1 is not an accumulation…

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