13. Which of the following sets of vectors DOES NOT span R*? {{)() (} {()0 {O ()0} {()) (} (a) 3. (b) (c) (d) (e) None of the above - they all span R3 (3 1 1 2 0 0 -2 0 0 -1 1 14. Let A = The mumber of free variables in the linear system Ar = 0 is (a) 0 (b) 1 (c) 2 (d) 3 (e) None of the above {() (O} - md B - {() ()} « 15. Consider the two bases B = of R2. What is the transition matrix from B, to B2? (b) 6 ) (G ) (c) (d) (e) None of the above

Q: 2. Let T: C² C³ be defined by T((x, y)) = (x,x+y,y) and S: C³ → C² be →> defined by S((x, y, z)) =…

A:

Q: The cantilever beam shown in Figure 2.1 is subjected to linearly distributed loads. a) Draw all…

A: From the figure, we can see that the linearly distributed load is w(x) = 120 kN/m and the length of…

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

A:

Q: 6. Let p(t) = ² + 1 and g(t) = t - 4 be vectors in P2 with inner product (f(t), g(t)) = f(t)g(t) dt.…

A:

Q: The Wronskian of the vector functions X1 e', X2 = 3 e2t, X3 0. O W (X1, X2, X3) = 55e6t O W (X1, X2,…

A:

Q: 2) Use Green's Theorem to compute f F df where F= (3x +4y)i + 2xyj and C is the triangular path from…

A:

Q: a) (2. 0. 1) b) (3. I. -1)/ 3 (0, 1.0 )(3,7. 5) [1] ) (18. 43, 30) S Q9. Let the vectors x1, Xz and…

A: Answer

Q: Example 7. Show that T: E2 → E2, defined by is not linear T((x₁, x₂)) = (x₁ + x2, x1 - x₂ +1)

A: as we now for a linear map T(01 ) = 02  where 01 is zero of domain and 02 is zero of codomain

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: 4. Let ci (1,1,1,1), c = (2,2 + x, 2,2), c = (3,3,3 + x, 3) and c✈ = (3, 4, 4, 4+ x). Find all x €…

A: The vectors {c1,c2,c3,c4} will be linearly dependent if any one vector is scalar multiple of the…

Q: Let B = {1, t − 8, (t − 8)²} : - and C = {1, t − 5, (t − 5)²} - be two bases for P2. The function…

A:

Q: Show that the system x' =Ax has constant solutions otherthan x(t)= 0 if and only if there exists a…

A: Given data : x’=Ax  we have to prove that system x’=Ax has constant solution other than x(t)=0 if…

Q: 1. Show that the given vector functions are linearly independent on (-∞, ∞). X₁ (t) = = 0 X₂ (t) H…

A:

Q: Let V be a general vector space and u, v € V be linearly independent. (a) Show that the vectors x =…

A:

Q: 7). Given functions F(x, y) = (x²y x³ – y + 4, ) and G(u, v) = (u² + v, v) compute the Jacobian…

A: We have to compute the Jacobian derivative matrix of function G o F at the point (1.0).

Q: fi(x) 1 A = {fi, f2, f3} c F(R) and f(x) = f:(x) = |x+ 1| PROVE: A is linearly independent. (A L.I.)

A:

Q: Determine the Jacobian matrix of the function f:R³ → R³ given by f(x,y,z) = (3x²y+ yx²,4x²yz², 6xz³…

A:

Q: Let y 1 = e3x and y 2 = xe3* . Which of the following statements is true about W(y 1, y 2) ? (a) W(y…

A:

Q: 42. Let M be 2x2 matrix with real entries. Consider the linear system of ODE given in the vector…

A:

Q: Let A(t) = f(t)i + f₂(t)) and B(t) = g(t)i+g₂(t)), te[0, 1] where fi, f2, 81, 82 are continuous fun-…

A: Solution

Q: If Q : R" → R is a quadratic form, then Q(x) attains its maximum and minimum values on ||x|| = 1 at…

A: Given statement is that if Q:ℝn→ℝ is a quadratic form, thenQ(x) attains its maximum and minimum…

Q: 5. Let S {V₁, V₂, ₁), n 22 be a set of nonzero vectors of V. Each of the following statements is…

A:

Q: 2. Let T: P2(R) → R³ and 4 2 [T]; 1 1 1 where B = {1, x, x²} and y = Is T invertible? {(1,0,0), (0,…

A: The solution is given as

Q: 2) a) Let T: C² C²: Ty = [x+2y-iz] 2y + iz Lix+z-2z Find [7] B [7] B and P where

A: Given information  T  :  C2 →C2T xyz = x+2y-iz2y+izix+z-2z simplify it, T xyz = x+2y-iz2y+izix+z-2z…

Q: 9. In this exercise, T: R2 R2 is a function. For each of the following parts, state why T is not…

A: Given function is T:R22 is function we have to prove that why T is not linear in upcoming parts.

Q: {(1, 2; 3 ), ii) {(1, 1, 1). (0, 1, 1). (0, 1, –1)}. 0. 1). (2, 1, 3)} spar

A:

Q: 1. Let A=M³₂ (f) be the matrix B' associated with the linear map f(x,y,z) = (3x − y +…

A:

Q: 1. Let A=MB.(f) be the matrix associated with the linear map f(x,y.z) = (3x-y + 4z.y+z-x.x+y+z)…

A:

Q: 9. Let L: R? → R² be the linear operator given by the rule L(r,y) (x + 2y, 3.r + 2y) . Find an…

A:

Q: 4) Find the Hermitian form [18] [2] X = H= XTAX where

A:

Q: Define T : P4(R) → P4(R) by T(p)= ((3x^2 −4x+9)p)'' (a)  Show that T is a surjective linear map.…

A: It is given that T:P4(ℝ)→P4(ℝ) defined as T(p)=((3x2-4x+9)p)'' for p∈P4(ℝ)Basis for P4(ℝ) is {1, x,…

Q: Let q be a quadratic form on R, defined below with u = (r1, 12, 13). Find the rank and the signature…

A:

Q: 3) Calculate Wronskian and evaluate for which values a, b, c e R the following sets of functions are…

A:

Q: Suppose T ∈ L(V ), m is a positive integer, and v ∈ V is such that Tm-1v does not equal 0 but Tmv=0.…

A:

Q: T is a linear operator on V, (R). B, = {(1, 1, 0), (1,0, 1), (0, 1, 1)} and B, = {(1, 0, 0), (0, 1,…

A:

Q: 1.3.2 Determine which of the following functions are and are not linear forms. *(a) The function f :…

A:

Q: Let y = 10 U₁ 11 -- and W = Span{u₁} a. Let U be the 2x1 matrix whose only column is u₁. Compute UTU…

A:

Q: (b) By determining constants c,, C2, C3 and c4, which are not all zero and are such that cifi(x) +…

A:

Q: 7. Let V={(x,y) x,y € R}. Suppose addition and scalar multiplication are defined using the following…

A:

Q: 7. Determine if the set S = {-1+x², 5 + 2x} of vectors in P2 orthogonal with (f. 9) = f(r)g(x) dr.…

A: The given set,    S=-1+x2, 5+2x