let P be the linear space of all polynomials of degree A. 1 B.n-1 C.n+1 D. n. E. None of the above

Q: For an irreducible polynomial p(X) = 1+X + X² € F[X] consider F4 generated by p(X). Find x1, x2 E F,…

A: Sol

Q: Let T: P4(R) → P4(R) be defined by T(f(x)) = xf"(x). Find a Jordan canonical form of T.

A:

Q: b. S = {1+t,1– t,t – t²} in P2 using the inner product (p(t), q(t)) = | p(t)q(t)dt (P2 is the set of…

A:

Q: 3. Let L be a linear operator on R and let L(1) = a . Show that L(x) = ax for all x E R .

A:

Q: Determine the kernel and range of each of thefollowing linear operators on R3: L (x) = (x1, x1, x1)T

A: It is given that,

Q: Proposition 6. Consider the function g : Z² → Z² defined by g(m, n) = (m - function g is a…

A: Consider the function , g : ℤ2 → ℤ2 defined by ,  g(m,n) = (m - n , n - 2m). We need to show that ,…

Q: Let T₁, T₂, be the following bounded linear operators ¹ T₁(x₁, x2. X3,) = (x₁, X₁, X₁, X₁, ...)…

A: Here, in the question, there are given the bounded linear operators l1→l∞ which are given as…

Q: 1. For each map 0; below, either prove the map is linear or show that it is not linear. a lal a. 01…

A: Answer

Q: 9 T1, T2 are normal operators on an laner produce space with the property that either commutes with…

A:

Q: Find the Wronskian of the given pair of functions: O x² et O et O xet X O² x, xet

A:

Q: Let T(x1,x2)=(2x1-2x2,9x1-8x2). SHow that T is invertible, and find a formula for T^-1(x

A:

Q: Find the Wronskian of the given pair of functions: X O x² et O xet i, xet

A:

Q: Let f(T) and g(T) be polynomials in Q[T] whose gcd in Q[T] is 1. Show that 1 is also their gcd in…

A:

Q: then L is a linear operator. Let L(x). R R be a mapping such that L( X2 Select one: O True O False

A: We will check whether the given statement is true or false.

Q: [M] Determine whether w is in the column space of A, the null space of A, or both, where -8 5 -2 2…

A: Consider the provided matrices, First find the product of A and W. Here, product of A and W is…

Q: Let k e Z+. Let A be a set such that |A| = k + 1, and let a e A (i.e., a is a fixed but unspecified…

A:

Q: Let What is the rank of AT A? Answer: A 3 0 0 0 0 0 5 11 −4 0 3 6 5 2 0 0 3 6 2 0 0 0 8 0 0 00 0 0

A:

Q: Let f : A → B and g : B → C be bijections. Prove that gof is invertible and that (g o f)-1 = f-1 o…

A: Problem based on function.

Q: The set of polynomials {1, x² + 3x“, 2x² – 7x4} is a linearly independent set in P4.

A: NOTE: Due to time limit answer of 1 is given here, for other please ask in a different question.…

Q: 2.54 For each n e N, let X„ be the set of polynomials of degree equal to n, and let Pn = U-o Xk.…

A: Sol

Q: Where x = (X1, X2) E R². 3- N: R2 - → R defined by N(x) = /x1² + x2² is a norm on R². (x1, x2) E R².…

A: Let X=(x1,X2)€R^2

Q: Let T: R" → R" be defined as T(x₁,x₂,...,xn) = (x₂ −X₁, X3 X₂, ..., x₁ - xn) of T Discuss the…

A: The matrix is invertible if the determinant of the matrix is non-zero. The matrix is diagonalizable…

Q: Let x = P[a,b] be the set of all polynomials over [a, b] with subnorm 11.11.00. Using Uniform.…

A: Given that: X=Pa,b be the set of all polynomials are a,b with suspension ∥·∥∞. The objective is to…

Q: A. Let R be a relation on set A. A={x : x E J and 0sxs 12}, and R={(a,b) : la - b| is multiple of 4}…

A: There are two questions given and it is discrete math. The solution to the above questions is given…

Q: Let L be a linear operator on R1 and let a = L (1). Show that L (x) = ax for all x ∈ R1.

A: It's a highly simple question.

Q: 44. Let A = {-2x+3 x € R} be a bounded subset of R. Find sup A. A B. - C. 1+ v2 D. 1- v2 E. None of…

A: First try is finding maximum  Luckily we got maximum therefore that became as an supreme

Q: Find bases for the null and range spaces of 1 2 3 1 1 1 A = 3 4 5

A:

Q: Determine the kernel and range of each of thefollowing linear operators on R3: L (x) = (x1, x2, 0)T

A:

Q: Let f(x) = + 3x? + 4x. a) What is the kernel of f? b) Compute the minimum and the maximum of f.

A:

Q: Let S be a set of nonzero polynomials in P(F) such that no two have the same degree. Prove that S is…

A: Let S be a set of nonzero polynomials in P(F) such that no two have the same degree.

Q: *** Let R be a unique factorization domain. Let p E R. If x is irreducible, then it is prime.***

A:

Q: 9. Let F : R" → R". Prove that for each a the mapping h→ dF[a; h] is linear on R".

A: Given: F:ℝn→ℝm   To prove: For each a, the mapping h↦dFa;h is linear on ℝn

Q: The space 7P is a Hilbert space when p+2 0 p=3 0 p#-2 O p=2 0

A: When p=2

Q: Let T be the linear thansformation from R5 to Rª defined by X1 1] X2 X1 1 3 2 X2 -8 1 -2 2 0 -1 4 X3…

A:

Q: Let A = Z[r]deg<n_denote the set of all polynomials in x with degree less than or equal to n. Show…

A: We will prove this result by first giving a bijection from Z^(n+1) to Z_n[x]  where, Z^(n+1)=Z×…

Q: 3. (a) Show that the space P of all polynomials p(t) with real coefficients is not complete with the…

A: Given that, ℙ be the space of all pt polynomials with real coefficients with the norm p=sup0≤t≤1pt…

Q: A. Let R be a relation on set A. A={x:xeJ and 0 SxS 12), and R={(a,b): a-b is multiple of 2)…

A:

Q: . Let a := V1+ v3. Find the minimal polynomial of a over Q.

A:

Q: Let T be the linear thansformation from R³ to R4 defined by 1 -8 1 X1 1 3 2 X2 X2 -2 2 4 X3 X3 3 0…

A:

Q: 2 2) Find a number t such that -3 is not linearly independent in R.

A:

Q: 3. Find the generator of X = Xo + ut + oWt, where W is a standard BM, and identify the PDE satisfied…

A:

Q: b) For 7 in R", we say that is in the kernel of T if T(x) = 0. If x₁ and 2 are both in the kernel of…

A:

Q: Let a ∈ G be such that |a| = ∞ . For m≠n prove that a^m ≠a^n

A:

Q: Consider the polynomials p,(t)=7+t and p2(t) = 7-t. Is {p1, P2} a linearly independent set in P,?…

A:

Q: Find a formula for the Hardy-Littlewood maximal function of the characteristic function of [0,1]U…

A: Given: The formula is determined by using the Hardy–Little wood maximal function of the…

Q: Let a; b; c 2 N such that a + b = c. Prove that gcd(a; b) = gcd(a; c) = gcd(b; c).

A: The given problem is to prove the following condition for given equation a+b=c, where a,b,c belongs…

Q: Let K„(x) = ane¯nl2l, x E [-T, T]. Prove that there are constants a, for which the K, form a family…

A: Given  Knx=ane-nx , x∈-π,π Let Kn form a family of good kernels. Then…

Q: Determine the largest subset S of C where the function ƒ(z)=3x² + 2x−3y² −1+i(6xy+2y) is analytic;…

A:

Q: Let f, g, and h be linearly independent functions defined for all real numbers, and construct three…

A:

Q: Show that The Hermite polynomials can be difined by the integral representation H, (x) = (-i)"2"e**…

A:

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.