Suppose that T :C C is R-linear. Show that T is C-linear if and if T(iz) = iTz for all z.

Q: Let w = r = uv, s = u – 2v. Use appropriate forms of the p2+ s2" dw dw and du chain rule to find in…

A:

Q: Show that ex and e−x are linearly independent in C(−∞,∞).

A:

Q: Determine the Linearity of this DE. * ay dy ニ X Linear O Non-Linear

A:

Q: Define: T:P, → R by T(P) = [ p(x)dx. What is the kernel of T ?

A: Given that: T:P2→RTp=∫01px dx

Q: Is {x² – 2.x, x²+x, x²} linearly independent dependent? Prove your answer. or

A: Answer

Q: The mapping $ (y) = fy' : H→H* defined by is not a linear m

A:

Q: Suppose uy, v2, Vm is linearly independent in V and w E V. Prove that dim span (v1 + w, vz + ,... "m…

A:

Q: The functions y1, Y2, . .. , Yn are linearly independent, if and only if W (y1, 32, ..., Yn)(to) #0…

A: If two functions are linearly dependent then the Wronskian is zero for every value of t.

Q: please solve this linear algebra problem.

A:

Q: { (0,3), (0,1) is linearly independent in R? ylgn O Ins

A: The given vectors are  (0, 3) and (0, 1) To find whether they are linearly dependent or not

Q: Suppose b, c E R. Define T: R³ → R² by T(x, y, z) = (2x − 4y + 3z + b, 6x + cxyz). Show that T is…

A:

Q: Suppose u1, ūz and ūz are linearly independent. Let īj = ủ1+ ủ2, 02 = ủ1 + ủ3 and ī3 = ū2 + 2ūz.…

A: Given: Let u1→,u2→,u3→ are linearly independant. Let v1,→=u1→+u2→,v2,→=u1→+u3→,v3→=u2→+2u3→

Q: The set of functions {Vx, x, x²} is linearly dependent on (0, oo). True False

A:

Q: find the linearization at x = a. f (x) = x4, a = 1

A: Given function: Using the formula of linearization of a function f(x) at x =a :

Q: 2 Given t = -1 and U = -1|, for which is the set {u, T, w} {d, v, 2 linearly independent?

A: Given that, u→=1-12 and v→=2-11 We have to find the vector w→ for which the set u→, v→, w→ is…

Q: The functions y1, 2, ..., Yn are linearly independent, if and only if W (91, 32, ..., Yn)(to) #0 for…

A:

Q: Use the generalized chain rule Question of Let f(x, y, z) = 2x(y+ z), x = uv, y = v + w, and z = w².…

A:

Q: The linearization at a = 0 to v4 + 4x is A + Bx. Compute A and B. A = B = %3D II

A:

Q: if u=f(x,y), x=g(w) and y=h(w), using chain rule, what is ∂u/∂w?

A: Silution : Given that u = f(x,y)

Q: = In z are two linearly indepen- of ry" + y' = 0, then the partic- - ry" + y' = f(x) can be written…

A: We have to choose the correct answer from the given options according to the given condition

Q: If z = f(u) + g(u), where u = = and v = x · y. Prove that x · Zx – y · Zy = y² ·Zyy – x².Zxx using…

A:

Q: 4) Suppose that T : R³ → R² is defined by T((x, y, z)) = (1+2x – z , 3 – x + 6y) Is T linear?…

A:

Q: Show that the functions e^x , e^2x , e^3x are linearly independent.

A:

Q: The functions y1, Y2, . .. , Yn are linearly independent, if and only if W (y1, Y2, ..., Yn)(to) # 0…

A: We can use the Wronskian test to check the functions are linearly independent or dependent.

Q: Consider x1 = et and x2 = v(t)et. If x1 and x2 is linearly independent, then v′(t) = 0. Is this…

A:

Q: Show that the function f(x, y, 2) = (x² + y² + 2e**)4 is strictly convex on R³.

A:

Q: Find the domain of R. R(t) In(1 t) √t+1, e-¹/|t| {(VEFT, 0-10m, bu(1-2)}) t = (1,0,-1) " , if t #0…

A:

Q: 1 if i= j 0 otherwise. P:(c;) = 5 is a linearly independent set.

A:

Q: 1. If x, y, z E R, prove that (c) ||x| – |y|| < |x –yl. (d) \xy| < }(x² + y²)

A:

Q: Suppose v1, ..., Vm is linearly independent in V and w e V. Prove that if vi + w, ..., Vm +w is…

A:

Q: 2. Use Wronskian to show that fi = x, f2 - Inz, and fa C (R). z Inz are linearly independent in

A: First we will calculate the wronskian of given functions f_1,f_2,f_3

Q: Consider the Simple Linear Model: 4 = Bo + Biz, + e, where i = 1,...,n, and e ~ IID(0, o?). Suppose…

A: Simple linear regression model: A simple linear regression model is given as y = b0 + bx, where y is…

Q: M1 := sup ||T(x)|| and ||x||=1 M2 := inf{C > 0 : ||T(x)|| < C||x|| for all x e R"}. a) Prove that M1…

A:

Q: 3. Let T: M₂2 → R² be defined as T ( [])=(a - b, cd). Determine whether or not I' is a linear…

A: I have given the answer in the next step. Hope you understand that

Q: 2 Given u -1 and U , for which w is the set {u, v, w} linearly independent?

A:

Q: 6. If u = given by x = +2, where x+y+z= 1, prove that the stationary value of u is y b a+b+c' a…

A:

Q: suppose that x,y,z∈R3 are such that x.(y×z)≠0. prove that x,y,z are linearly independent.

A:

Q: Let, T: R3 → R³;T(x, y, z) = (2x + y, y – z,2y + 4z) Find the rank and nullity of T.

A:

Q: Suppose vi, v2, ..., m is linearly independent in V and w ɛ V. Show that v1, vz, ..., Um, W is…

A: v1, v2,......., vm is linearly independent in V and w∈V

Q: The functions y1, Y2, · . . , Yn are linearly independent, if and only if W (y1, 22, *. Yn) (to) # 0…

A: Consider the following differential equation, y''+pxy'+qxy=0

Q: If r(t) = (V10 – t, In(t – 1) - then the domain of r is

A: Domain = (1, 10]

Q: 3. Determine whether the functions e and re are linearly independent or dependent by calculating…

A: f= {e^2x , x.e^2x} is Linearly independent if Wronskian is non zero.

Q: find the linearization at x = a. y = e√x , a = 1

A: Given function: Using the formula of linearization:

Q: In C[0, 1], are the functions {x2 , x3 , e2x} linearly independent?

A: Linearly dependent and independent functions: Two differential functions f(x) and g(x) are said to…

Q: 41. Let f(x, y, z) =x?y+ y²z. Use the Chain Rule to calculate af/as and af/at (in terms of s and f),…

A: Since f is function of x, y and z.  For partial derivative f with respect to s and t we using chain…

Q: Suppose that T: R2→ R2 is linear, T(1, 0) = (1, 4), and T(1, 1) = (2, 5). What is T(2,3)? Is T…

A: Given,

Q: Let f(x, y, z) = x2y + y2z. Use the Chain Rule to calculate af /as and af /at (in terms of sand t),…

A:

Q: find the ideal.Sdution for the following Linear model: min(z) =3X+2 メ、+Xレ26 2 Xi+2 XxZ12 メト→メ20

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.