(a) Show that T is linear and bounded, and find ||T||. (b) Show that T is bijective and find T-'.

Q: (a) Define the T1-space.

A: Define the T1 - space

Q: (a) Prove the set (A(*)./a(*),fa(a)) is linearly independent. (b) Use Gram-Schmit process to find an…

A: Given,     f1(x)=3,  f2(x)=1-x, and f3(x)=x2+2x-1

Q: 18. Determine whether each of these functions is a bijec- tion from R to R. a) f(x) = -3x + 4 b)…

A: As per guidelines, we will solve first three subparts of the question. a) Given function is fx=-3x+4…

Q: (-х+а, х < —1 If f(x) = {x² + 5, –1 < x < 2 is continuous then х — b, х 2 2 a + b =?

A:

Q: 3. Solve the initial value problem for a 2 x 2 system of linear ODEs for r(t) and y(t): (3 = Use the…

A: In this question we will solve the given equation by using elimination method.

Q: Prove that f(x) = 5x2 + 3x – 26 is θ(x2)

A: Given: f(x) = 5x2 + 3x – 26 is θ(x2)

Q: positive real numbers such tha ith proof the minimum value of

A: Given, x,y,z be a positive real number such that x4+y4+z4=1.The minimum value of…

Q: Show that V spans P² where V = {1 + 2t + 2t², t + t², 1 − t²}

A:

Q: Consider the interpolating polynomial for f(x) = x³ based on th points (0,0), (1,1) and (2,8). Find…

A:

Q: Find the locus of the points z that satisfies 3(22 + z?) + (8 – 16i)z + (8+ 16i)z – 10|z|? = 16.

A:

Q: Let f(z) I+bI has a minimum value at the point (9, 6). Then the value of b is %3D Select one: O 4 0…

A:

Q: Prove that the quartic function q(z) = r* - 16 has only

A: Given that: q(x)=x4-16

Q: Let vị = V2 = V3 V4 and -[1} Confirm that p = V1 +v2 + ¿v3 + ¿V4 and vị – V2 + V3 – V4 = 0. Jse the…

A:

Q: b) Compute Re fli) when : f(2)= √Z - 1² and f(0) = -i c) Which of the following functions are…

A: Analytic function : A function f is analytic at a point complex point z=a iff the f(a) is a finite…

Q: a) f(AUB) = f(A) U ƒ(B). b) (i) f(An B) ≤ f(A)n ƒ(B). (ii) Find a function f {1,2} → {a}, and A, B C…

A: a)  We  have  to prove  that  Now,      Thus,  (Proved)[since, b) i) We  have  to show  that Let  be…

Q: (d) Show that T is bounded and find ||T||. (e) Find the adjoint T* of T.

A: We will use the vector/matrix notations to visualise things easily.

Q: , Consider the interpolating polynomial for f(x) =x² based on th points (0,0). (1,1) and (2,8). Find…

A:

Q: a) Prove the one point rule - 3-version: (3x)(x = t ^ A) = A[x = t] if x is not free in t. b) Prove…

A:

Q: Consider the function ƒ (x, y, z) = 2x² + 5y² + z². a) If ƒ (x, y, z) is the profit made from…

A:

Q: Prove that the linear function f(x) = ax+b is continuous.

A: To prove that the given linear function is continuous.

Q: Show that f: R\{2} → R\{5} defined by f(x) = (5x+1)/(x - 2) is bijective.

A:

Q: Determine whether the following set of polynomials is linearly independent (1 – t)?, t – 2t2 + t³,…

A:

Q: Q4) Determine whether the set S is linearly independent or linearly dependent a) S = {(2,1,–2),…

A:

Q: Prove that the function : R-{1} →R- {2} defined by 2a f(x)= s bijective. 2 1

A:

Q: Let φ1(x) = 1, φ2(x) = x/|x|, φ3(x) = x^2 − 1/3 for −1 ≤x ≤1. Show that (φ1,φ2,φ3) form an…

A:

Q: Let T: R3 -&gt; R2 be defined by T(a, b, c) = (a + b, 2a —- c). DescribeT-1(1, 11).

A: T: R3 -&gt; R2 Hence, T-1 : R2 ----&gt; R3 So, necessarily we have to find a, b &amp; c such that…

Q: Determine whether or not T = (6,-3) belongs to the null space of A' if A 1

A: the answer is

Q: b) Let P2(R) be the polynomial space of all polynomials of degree less than or equal to 2 whose…

A: To find out which of the following sets are linearly independent .

Q: Consider the function f(x, y) = x^2 − y^2 over the square centered at 0 whose length of each side is…

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: 1 -1 X%3D X. 0 1

A: Given: X'=1-10 1X Let A=1-10 1 Since A is a upper triangular matrix . So, its eigenvalues are just…

Q: Verify that ||v-w||^2=1 when ||v||=2, ||w||=1, and v*w=2

A: By the definition of inner product and norm we can derive the following formula : ‖u-v‖2=⟨u-v,u-v⟩…

Q: (2) Let f(x) = ax³ + bx? + cr + d, where a, b, c, d e R and a + 0. Prove that f(x) must have a root.…

A:

Q: — 2х1 — 5х2 + 8хз x1+3x2 – 5x3 +x4 |3x1 + 11х2 — 19хз + 7x4 X +7x2 — 13хз + 5х4 x1 X2 . Let T X3 X4…

A: The given transformation T can be written as            Tx1x2x3x4=-2-58013-51311-19717-135x1x2x3x4is…

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

A:

Q: Let X₁ = [1 3 −1 1]²‚×₂ = [3 −1 3 1]′, and x3 = [-1 2 −3 1]. Is b = T an element of span{X₁, X₂,…

A: No, we can write it.