Is the set W={ f(x)e P(F): f(x)=0 or f(x) has degree n } a subspace of P(F) if n21? Justify your answer.

Q: 4. The set W of polynomials of the form a² + α₁x + α₂x² + 3x³ + α4x4, where a₁ = a₂ and a₁ = az is a…

A: 4. Given that the set W is of the polynomials a0+a1 x+a2 x2+a3 x3+a4 x4 where a0=a2 and a1=a3 is a…

Q: Consider the subspace V of R4 given by 1 3 1 V = span || -1 1 1

A: Solution:-

Q: Consider L = I – A with A : X → X is a compact linear operator in a normed space X. Show that the…

A: The given problem is related with functional analysis. Given that L = I - A , where A : X → X is a…

Q: Let W = {a + bx + cx² + dx³| a + b = 0,c – a = 0, and d – 3a = 0 } be a subspace of P3. Then the…

A: dimW = 1

Q: Let A = { | 2.x |x € R}. Is A a subspace of R³? З Let B x + y + z = 0}. Is B a subspace of R³?

A:

Q: 10. Under the usual operations in R2, which of the following subsets is a subspace? (a) y = 2ar…

A:

Q: a. . Is the set of vectors of the form a subspace of R³? Show your proof. La+2b

A:

Q: If V is a linear subspace of R", and x1 , ×2 , ... , Xn E V, then we must have span {x1, X2, Xn} =V.…

A: To determine that the statement, if V is a linear subspace of ℝn and x1, x2,....,xn∈V, then spanx1,…

Q: Q3 shew that W2 = subspace of R2 is a

A: To show any subset of vector space is a subspace , prove the following condition : Set is closed…

Q: 1. Determine whether w - {[:]•r:a-4-0} X1 E R? : : x1 – 23 = X2 is a subspace of R?.

A:

Q: Produce an example of an element of Co (R, R) which is not an element of the following vector…

A: Correct answer is f(x)= Sin(x) which is infinitely many times continuously differentiable but it is…

Q: For a certain subspace W, the orthogonal projection of the vector y = 3 onto W is ŷ 2. Use this…

A:

Q: Question 3. Are the following equalities of subspaces in P3 true or false? Justify your answer. (1)…

A:

Q: Prove that the set of mxn upper triangular matrices is a subspace of Mmxn(F).

A: We have to prove that -

Q: If Sz is a subspace of R" of dimension 3, then there cannot exist a subspace S, of R* such that Si c…

A: This staement is true.

Q: Verify that V5 is a subspace of RS

A: Please check step 2 for solution.!

Q: 4. Let S be the subspace of P₂ defined by S = span{1-x, 1+x, 5, 1-x², 2x}. Find dim (S), and show…

A:

Q: 4) Let X, Y, Z be subspaces of a vector space V and assume that Y CX. Prove that Xn(Y+Z)% =Y+(XnZ).…

A: 4. Given The modular law of sub-space: Let X,Y,Z be subspaces of a vector space V and assume that…

Q: If Sz is a subspace of R* of dimension 3, then there cannot exist a subspace S2 of R* such that S, c…

A: S1, S2 are subspaces of R4 such that S1⊂S2⊂R4 with S1≠S2 and S2≠R4.

Q: Let W = {a + bx + cx2 + dx'|c-3d = 0} be a subspace of P, Then the dimension of W is equal to %3D 4.…

A:

Q: b) Is W = {f(x):f is a polynomial with degree <3,f (0) 0} a subspace of P: all 3 ? Explain your…

A: To see if the given subset is a subspace of the vector space of all polynomials of degree less than…

Q: What is the dimension of w = {A € M2x2 (R) : A"| =ö} ĄT -2 1 2 This is not a subspace

A: W=AEM2×2(R) : AT1-2=→0  Let A=abcd, a,b,c,dER     AT=acbd     AT1-2=→0  ⇒acbd 1-2 =00 ⇒a-2cb-2d=00…

Q: 4. The set W of polynomials of the form α₁ + α₁x + α₂x² + α3x³ + axª, where a = a₂ and a₁ = az is a…

A:

Q: Let W = {a + bx + cx? + dx³|c – 3d = 0} be a subspace of P3. Then the dimension of W is equal to…

A:

Q: The set {(x,3): x is in R)} is a subspace of R2

A: The given set is x,3: x is in R. It is known that any subspace should contain identity element.

Q: Suppose that U cr|xyer F4 y EF, U is a subspace of F4- you don't have to! Find a subspace V of F4…

A:

Q: Is M = {(x,y,z)|x is nonnegative} a subspace of R³? Yes or No? Justify or give/find a…

A: Given  M =  x, y, z  x is nonnegative

Q: 4. Show/prove whether the following is a subspace. 1 {x € R3:x =| 3| +s 2+t1,for s,t e R}

A:

Q: Find a basis and the dimension of the subspace W of P3= {ax3+ bx2 + cx + d} defined by W = {PEP3:…

A:

Q: In C[−π, π], find the dimension of the subspace spanned by 1, cos 2x, cos2 x.

A:

Q: (b) Define subspace and Verify whether is subspace v = {(x,y,2):x,y,2ER² and x-2y+3z = 5}

A:

Q: 3. Is the set of all vectors (x, y) in R2 with the usual addition and scalar multiplication, a…

A:

Q: Is the set of all differentiable real-valued functions defined on R a subspace of C(R)? Justify your…

A: Given:Is the set of all differentiable real-valued functions defined on R a subspace of C(R)?

Q: Is the set of noninvertible 2 × 2 matrices a subspace of R2×2 ?

A: Given, ℝ2×2=abcd: a,b,c,d∈ℝ Let M be the set of all 2×2 noninvertible matrices  i.e.…

Q: 12. Let A be a fixed n x n matrix and let S = {B e Mnxn|AB = BA}. Is S a subspace of Mnxn?

A:

Q: The set of all points in R3 satisfying x + y - z = 0 is a subspace. Note that the set of point…

A: It is known that any subspace of ℝ3 should contain an additivities identity element 0, 0, 0.

Q: Example 3. The operator Ax = d/dt x(t), defined on the subspace M of C[0, 1] consisting of all…

A: consider the operator d/dt on C[0, 1] with domain consisting of continuously differentiable…

Q: pan(1,1 – x, 1 +x², 2x – 3) = span(2, 2 – x°, 2x² – x, x + 1). - %3D - -

A:

Q: 4. Let S be the subspace of P2 defined by S = span{1-x, 1+x, 5, 1-x², 2x}. Find dim(S), and show how…

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

A:

Q: Let W {a + bx + cx2 + dx³| a+b = 0 and c- 3d = 0} be a subspace of P3. %3D Then the dimension of W…

A:

Q: 4. The set W of polynomials of the form ao + α₁x + a₂x² + 3x³ + a₁x¹, where ao = a₂ and a₁ = a3 is a…

A: Set W is polynomial and W is subspace of P4. We find number of arbitrary constant in set W.

Q: and uz = Find the closest point to y = in the subspace W spanned by u1 = The closest point is

A:

Q: What is the dimension of the subspace 0 2 3 1 H = x1 x2 x3 X4 ER¹ : 2x1 3x2 + 4x3 5x4 = 0 ?

A:

Q: 4. The set W of polynomials of the form ao + a₁x + a₂x² + a3x³ + a4x¹, where ao = a₂ and a₁ = az…

A:

Q: If V is a 13 dimensional subspace of R20 then the dimension of V- must be

A: Definition of Orthogonal Complements Let V be a subspace of ℝn. Its orthogonal complement is the…

Q: The cosine space F3 contains all combinations y(x) = A cos x+ B cos 2x+C cos 3x. Find a basis for…

A:

Q: Let A e Mnxn and W be the set of all solutions for the homogenous linear system AX = 0. Prove or…

A: Given, A∈Mn×n and W be the set of all solutions for the homogenous linear system AX=0. We have to…

Q: (5) Suppose that T E L(V) is such that all subspaces of V of dimension dim(V) – 1 are T-invariant.…

A: Please check step 2 for the solution.!

Q: Let W = {a + bx + cx2 + dx³| a + b = 0 and c – 3d = 0 } be a subspace of P3. Then the dimension of W…

A:

Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.