3. Let V be a finite dimensional vector space and let (.,.,) be an inner product on V. Suppose that V ‡ {0}. Define || · ||: V → [0, ∞) by ||u|| = √√(u, u). So || · || is the norm associated with the inner product. Let T: V → V be a linear map. Suppose that ||Tu|| = ||u|| for all u € V. Prove that T is orthogonal.

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: Definition 1. Fix a scalar product (·, ·) in R" and c E R", d e R and denote by H the open…

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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 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: 7. Assume that Vi, i = 1,2 are vector spaces over F € {R, C} and (, );, i = 1, 2 is an inner product…

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Q: For a polynomial p ao + a₁x + a₂x² + + anx, let p denote the polynomial obtained from p by removing…

A: Note:  As you requested only Part C(I) so I answered here only that.

Q: 5. Let V be an inner product space over C. Show that for any v, w e V (v, (v, w)w) = | (v, w) |² and…

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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: 1. Show that ||x|| , ||x||||x||₂ are vector norms. 2. Show that if x and y are two vectors, then…

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Q: Let P denote the vector space of polynomials (of any degree) and E P → R lenote the evaluation map…

A: Please find the solution in next step

Q: Q2: Consider the following normed spaces (C([a, b]), ll·ll.), (C¹([a, b]), Il-ll..) and (C¹([a,b]),…

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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: Let x = (13, −20, 15), v1 = (1, −2, 3), and v2 = (−1, 1, 1). Then the set {v1, v2} is orthogonal. It…

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Q: V be a vector space and T: V → V be a linear map such that T² = T. Show that img(T – I) ≤ kerT.…

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Q: 1. Let V be an inner product space. Suppose f,g E V such that f -gg (a) Prove that 1I2 = |lf- gll2 +…

A: To prove the required norm property in an inner product space, under the given conditions

Q: Proposition 18. Norms are conver. Proof. Let || - || be a norm on a vector space V. Then for all x,…

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Q: Given any Hermitian space E with › E, if (f(x), x) = 0 for all x € E, th I

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Q: 3. Let S: U → V and T: V → W be linear transformations between finite dimen- sional vector spaces.…

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Q: Show that R³ is spanned by S = {(1,1,1), (1, –1, 0), (0, 1, – 1)}.

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Q: 2. Suppose that V and W are vector spaces over F. Consider the cartesian product V x W, with vector…

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Q: Af a, B are two vectors in an inner product space V(F) and a, be F then prove that Re (a, B) = || a…

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Q: The value of the limit lim t-0 vectors in space, is (i) i+j+ k (ii) 2i+j+ k (iii) 2i - 2j + k (iv)…

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Q: Let V be a vector space and suppose there exist maps S, T = L(V, V) such that range(S) ℃ null(T).…

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Q: a normed space, mapping such that ker(T) = {0}). Show that: defines a norm on E. any vector space…

A: Answer given below.

Q: Consider linear vector space c¹ ([a,b]) = {f: [a,b] → Rf and f are continous on [a, b]} ||f||0,1:=…

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Q: 5. Let P2(C) be the vector space of polynomials of degree less than or equal to 2 over C. Let C3 be…

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Q: 4. Show that the norm on l0 given by I| (11, .., In...) |lo= sup{|r;| : j= 1,2, ...} can not be…

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Q: Let X be the vector space of all ordered pa y = (n1, N2), · of real numbers. Show that norms on %3D…

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Q: Theorem 1. Let U end V be vector spaces over the fleld F and let T be a linear transformation from U…

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Q: 1. Let V = R[x] be the vector space (over R) of all polynomials in r with real coefficients. Define…

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Q: 2. Let E be a vector space. A function p: E- lf is called a seminorm it pirx)=iripx). for ll ond…

A: We need to find an example of non-zero seminorm on R2 which is not a norm.

Q: Let V be a vector space over F and w₁, W2, W3 € V. Suppose the list w₁, W2, W3 is linearly…

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Q: 1) Let V, W be F vector spaces. Recall that a skew-symmetric map is a function f: V × V → W such…

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