Problem 43. Let L: V → W be a linear function and define def Graph(L) {(v, L(v)) E V x W : vEV}. (a) Prove that Graph(L) is a vector subspace of V × W.

Q: 8.) What is the distance between an airplane flying at 120 feet above sea level and a turtle…

A: We find the distance between an airplane flying at 120 feet above sea level and a turtle swimming at…

Q: rove that W2 = {(a,b) E R² : a = 26²} is not a subspace of R?.

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Q: (b) If V = U OW is a direct sum of subspaces U, W C V then any element v E V can be written uniquely…

A: Given V=U⊕W is a direct sum of subspaces U,W⊂V.  Then, it is known that any element v∈V can be…

Q: 1. Let e Z(X) and let A = {xe X: f(x)=x}. Show that M is a subspace of X.

A:

Q: 7. Let T : V space W. Prove that the range of T is a subspace of W. → W be a linear transformation…

A:

Q: 8: Let v - (1, 1) and w (1, -1) belongs to R Then a vector u (a, h) e R' is in the R-linear span of…

A: linear span of v and w can be written as, t = xv+yw where, x and y are real numbers So,  t can be…

Q: 4. Let V1 = (-3) - () -- () 5 = V2 10 V3 = " 12 a) For what values of h is v3 in Span {V₁, V₂} ? b)…

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Q: 1 3 - -- () ---- () 5 V2 = 10 3 12 4. Let VI: a) For what values of his v3 in Span {V₁, V2} ? b) For…

A: The given vectors,.The objective is to find the values of  for which the vector  is in the  and  are…

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: be a linear operator on a vector space V(F). If = 0, t ce oft and null space of t. Also given an…

A:

Q: Find the closest point to y in the subspace W spanned by v₁ and v₂. y + 13 V₂ = 1 6 The closest…

A:

Q: (a) F = Q, V = R², and W = Z². (b) F = R, V = R², and W = {(x, y) E R? : ry > 0}. (c) F = R, V = R³,…

A: Let W≠0 be a subset of a vector space V over F with the same addition and scalar multiplication as…

Q: 6. Show that the closure Ỹ of a subspace Y of a normed space X is again a vector subspace.

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Q: Create a vector of from F(x,y,z) such that the x, y, &z components contain at least two variables…

A: Given : vector of the form F(x , y ,z)  To Find : vector of the form F(x , y ,z)  such that the x, y…

Q: Let V = P5(Q) the vector space consisting of polynomials in the variable x of degree at most 5. Let…

A: See below for soluton.

Q: Select true or false for each statement. 1. The vector b is in the kernel of T 2. The vector c is in…

A: Given,             And

Q: 9. For the linear operator LA R³ R³ defined by L₁(x) = Ax, for x column vectors in : R³, and 0 1 0 0…

A:

Q: 22. Let f be a linear functional on a subspace S of a finite-dimensional inner product space V. Let…

A: Please find the answer in next step

Q: 2. (а) ở and w are vectors in R? such that f(0) = w. Express f(27 – 3w) as a linear combination of i…

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Q: LA3-2. Suppose f is a nonzero linear function on some vector space V and U = ker f. Prove that V = U…

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Q: (3) Is Mmxn(Q) is a vector subspace of Mmxn (R) over R?

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Q: 1. Let V = C[-2, 2], the vector space of continuous functions on the closed interval [-2, 2). Let f,…

A: We can just multiply h and k and integrate  Then after simplifying we are getting answer

Q: If W, and W, are two subspaces of a vector space V (F) then prove that WinW, is also a subspace of V…

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Q: 1 (b) Find the span of vectors i = |0 and j = |1 in R3

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Q: II. Decide which of the following are subspaces of corresponding vector spaces: C R?

A: Vector space.

Q: then A(-2u - 20) = -2 ---B 5 and Au 3 Au = 2 4

A:

Q: Let E and F be two vector spaces with the same finite dimension dim(E) = dim(F) For every linear map…

A: Follow the steps.

Q: 5. Prove that for any vector x, we have (a) 1, <H<#H, (b)

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Q: or function By – k1 z) + ay (kz y + z) onal, then the values

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Q: 6. Let x#0 and y‡0. (a) If x 1 y, show that {x, y} is a linearly independ- ent set. (b) Extend the…

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Q: 3. Consider V = {a2x² +a1x + ao € P2(R) : a2 + a1 + ao = 0} . Use the subspace theorem to show that…

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Q: (ii) Let W = span{ } v = and The orthogonal projection of v onto W is projw(v) = and the component…

A: Given : v=-43-7    and W=span{ -2-32 , -3-1-2 } Aim: to find (a) projw(v)      ,and   (b) perpw(v)

Q: 6. Let T is a linear operator on a finite-dimensional vector space V. Prove that V R(T) + N(T)…

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.