Show that the set of real polynomials of n – degree (Pn) is a vector space with traditional addition and scalar multiplication

Q: Let S be the set of all ordered pairs of real numbers. Define scalar multiplication and addition on…

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Q: Show that the set  Fm×n of all  m×n  matrices over a field  F  is a vector space over   F with…

A: Given:- Fm×n: Set of all m×n matrices over field F . To prove: We have to show that the set Fm×n of…

Q: Let V denote the set of all m x n matrices with real entries; so V is a vector space over R by…

A: Here V is a vector space over the field of F if and only if for every x,y∈V and α,β∈Fvector αx+βy is…

Q: Determine whether the set, together with the indicated operation is a vector space. If it is not,…

A: We need to choose correct option.

Q: Let V be the vector space of set of all real polynomials over the field of real numbers ℝ. Let W be…

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Q: Let V denote the set of all differentiable real-valued functions defined on the real line. Prove…

A: Let the real line be the field. Suppose that f,g,h∈V be real valued differentiable functions. Sum:…

Q: Let V be the set of infinite sequences (a, ,a2,...) in a field K. Show that V is a vector space over…

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Q: Determine whether the set, together with the standard operations, is a vector space. If it is not,…

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Q: Give an example of a subset (not subspace) of R3 that has an infinite number of elements, is closed…

A: A subset W of vector space V over the scalar field K is a subspace of V if and only if following…

Q: Let V be the set of all polynomials over Q of degree 2 together withthe zero polynomial. Is V a…

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Q: Give examples of two DISTINCT bases for the vector space of polynomials with degree less than or…

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Q: (d) Show that the polynomials P₁ = 3 + 2x + x², P₂ = 2x+3x², p3= −2+x=2x² form a basis of the vector…

A: Using 1+2x-x^2=ap1+bp2+cp3 find a b c

Q: Find an orthonormal basis for the subspace of Euclidean 3-space below. W = {(x1, x2, x3): x1 + x2 +…

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Q: Let Z denote the set of all integers with addition defined in the usual way and define scalar…

A: We shall use the axiom A6 and A7 which contain the scalar multiplication to show this.

Q: Let V be a vector space over an infinite field. Prove that V is not theunion of finitely many proper…

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Q: Show that cc Lo is and o7 Scalars converg ing to a vector Subspace of Ľ is C,, the Space of all…

A: Let C be the set of all convergent sequences' C=an:an is convergent Define sequence addition an…

Q: Let P3 be the set of all polynomials of degree at most 3. That is P3 = {ao + at+ azt? + azt | ao,…

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Q: "Show that Q (rational numbers) is a vector space over R (real field). That is, a "vector" is now a…

A: To prove (V,+,.) is a vector space we have to prove that (V,+) is an Abelian group. Scalar…

Q: Let P, be the vector space of all polynomials of degree < 2. If W is the set of all polynomials of…

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Q: 6. Prove that an open interval (a, b) considered as a subspace of the real line is homeomorphic to…

A: An open interval a,b considered as a subspace of the real line is homeomorphic to the real line.

Q: Determine the dimension of the subspace of R consisting of all XeR* such that x1 + 2x, = 0 and X3 –…

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Q: Let V = {(a1,a2, ... ,an): ai ϵ C for i = 1,2, ... ,n}; so V is a vector space. Is V a vector space…

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Q: Prove that the dual of l° is not isometriç to, 1', but contains a proper subspace that is isometric…

A: Consider that there are bounded functions on l∞ which are not of the form fy=∑xkyk for some x. Let,…

Q: Let V be an n-dimensional vector space over F. What is the characteristic polynomial of the identity…

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Q: If V is infinite-dimensional and S-is an infinite-dimensional subspace, must the dimension of V/S be…

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Q: Let P, be the vector space of all polynomials of degree < 2. If W is the set of all polynomials of…

A: Let V be vector space for field F. A set W⊂V(F) is said to be subspace of vector space V if W is a…

Q: Prove that span (v1,v2,...,vn) is a valid subspace of R

A: We know that ,  spanv1 , v2 , .... , vn = a1v1 + a2v2 + ...... + anvn / a1 , a2 , ...... , an∈ℝ A…

Q: Let V be the set of all positive real numbers Determine whether V is a vector space with the given…

A: Given V be the set of all positive real numbers.  To determine whether V is a vector space with the…

Q: w that the space V = {(x1, x2, x3) ∈F3

A: To prove that the space V=(x1, x2, x3)∈F3|x1+2x2+2x3=0 forms a vector space.

Q: 3. The collection of all polynomials of degree exactly 3 forms a vector space 4. The collection of…

A: 3. False  Because null polynomial is not a polynomial of degree exactly 3.

Q: Consider the vector space P, of all polynomials of degree = a, a2+b, b2 , then the inner-product of…

A: We have to find the inner product of the following -  p = 2 + 1.7x q = 4.8 + 3.2x

Q: Let n € N and let V be the set of polynomials in the indeterminate x with coefficients in R of…

A: We will be solving part (b), (c), and (d) as mentioned. Given n∈ℕ. Now define:…

Q: (a) Suppose addition in R? adds an extra 1 to each component, so that (3, 1) + (5,0) = (9,2) in-…

A: As per our guide lines we can only answer one question and you would have to repost the other…

Q: (b) Show that the set FmXn of all m x n matrices over a field F is a vector space over F with…

A: here we try to satisfy all the conditions of vector spaces.

Q: Let Pn be the vector space of all polynomials of degree n or less in the variable æ. Let D² : P4 →…

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Q: If M is a closed subspace of a Hilbert space H then H %3DМӨМ'.Prove.

A: Complete inner product space is called Hilbert space

Q: Show that the set of all polynomials of degree at most 3 forms a subspace of Pn.

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Q: Prove that the dual space of 10 is not isometric to 11 but contains a subspace isometric to l1.

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Q: Find the dimension of the subspace Hof R2 spanned by

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Q: 1. Show that the set of all polynomials of deg=2 is not a vector space over reals. can this be…

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Q: Let W be an (n – 1)-dimensional subspace of V. Show that V has a basis B satisfying BnW = Ø. |

A: This question is related to vector space . Solution of this question is

Q: 2. Let P denote the vector space of all polynomials on R. Show that P is infinite- dimensional.

A: Let ℙ denote the vector space of all polynomials on ℝ .We have to show that ℙ is infinite…

Q: Determine whether the set, together with the standard operations, is a vector space. If it is not,…

A: Determine whether the set, together with the standard operations, is a vector space. The set of all…

Q: A real-valued function f defined on the real line is called an even function if  J( -t) = f (t) for…

A: Givenf(-t)=f(t)-even functions.To prove Set of even functions forms a vector space.

Q: 1. Show that the set of all polynomials of deg=2 is not a vector space over reals. can this be…

A: Answering part 4, as requested   Let f be a set of infinitely differentiable real functions. Given…

Q: The set of all polynomials of degree 5 under the standard addition and scalar multiplication…

A: here option (c) is true . because here field is not defined

Q: Let V be an n-dimensional vector space over the field of complex numbers C. Show that V over R has…

A: A complex number z can be written as z=x+iy.Assume a basis for V over C then extend it to V over R.

Q: 8. If Z is an (n−1)-dimensional subspace of an n-dimensional vector space X, show that Z is the null…

A: Given: n-1 dimensional subspace To find: Scalar multiple

Q: x=y=z

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.