a bonneled linear operator. Then prve. that. Kes's is clased. b) let 8:X>y be a

Q: The set of vectors in R? with y = -3x is a real vector space

A:

Q: Let T be function from R to R defined as T (z) = (r, x), %3D Then T is not linear operator O True…

A: FALSE  the detailed explanation is as follows below:

Q: Consider the following linear operator; T : R → R° given by T(x, y, z) = (y, x + z, z).

A: We have to find the real Eigen value of T

Q: Let J={a: aER}, then I is Borel algebra.

A: False Statement' Detailed Explanation

Q: Ex: © Suppise that the fiowing Aransferuntin Conneclğs the lei) and (ti) coordnate Syshems; a4ス?) 2…

A: Given that, ζ=x¯1=ex1+x2x¯2=ex1-x2 To find: a) Jacobian matrix J and J-1 b) Compute J¯

Q: Let L: R→ R be a linear operator, if L(X)= L(Y), then X= Y. Select one: O True O False

A: Given Linear operator is  L:R3→R3 we have to check if L(X)=L(Y) the X=Y

Q: If y is orthogonal to both and then y is orthogonal to every w in Span{u, v} V. O True False

A: We have to solve given problem:

Q: et T E L(V,W), and U e L(W, X). Prove that UT : V → X is also linear

A:

Q: det(A) Show that det(AB-') = %3D det(B)

A: Consider, Let A and B be the two n×n matrix, Now Consider, detAB-1 We know that the theorem, Let A…

Q: What is the dimension of the vector space S=abc detEM2x3(R): a=b=c=0}

A: Dimension of vector space is number of element in basis

Q: Find a norm of the operator A(x) = o (s4 + s? )x(t)dt in space C[0, 1]. a. 4 Ob. O c. 2 4 O d. 15 1…

A:

Q: Given that w = f (u, v) And u = xz. , V = dw = X az aw aw Prove that Z - ду

A:

Q: Use the definition of the dot product a b = |a||b| cos 0 to prove the Cauchy-Schwarz Inequality: |a…

A:

Q: what is the Null Space o %3D a)= T,:R3_ R³ (T,x= xxa 03 义.a 13D Txlt)= x' (t) (Differen tiateon…

A:

Q: Drag the postulate or theorem that proves the trieng AZWX AY XW a. ATSU AXVW # SSS E SAS

A: which criteria follow to congruent triangle

Q: Let E=RandT:R R be a Linear tiansformation I(,)=(2,3)and Tls0)=(3,2) Check whether T exists or noti2…

A: Given T:ℝ2→ℝ2 is a linear transformation such that T1,1=2,3 and T1,0=3,2. Now find the matrix of the…

Q: Let r(t) = ⟨t, t, t⟩. Compute r'(t) and interpret the result.

A: rt=t, t, t

Q: Let V be a vector space, and let T : V → M₂2 be linear. Let V₁, V₂ EV and suppose 2 T(v₁) = J 1…

A:

Q: Let w = r = Uv, s = u - 3v. Use appropriate forms of the + s²' dw dw chain rule to find and in terms…

A:

Q: Let X; ~ X"(1) i= 1,2, . - ·, n and assume they all uidependent are Y = Ź Xi %3D しニ」 ان منتتفينتمنل…

A: Given, Xi~χ12 ,i=1,2,...,n a. Y=∑Xii=1n Therefore, X1+X2+.....+Xn~χn2 above is possible using…

Q: Suppose V is a vector space and S, T = L(V, V) are such that Range Scker T=null T. Prove that (ST)²…

A:

Q: Let X and Y be normed linear spaces, and T E B(X,Y). Prove that 38 > 0 such that 8||x|| < ||T(æ)||,…

A: Consider the given information.

Q: If S and T are normal linear operators satisfying ST* = T* S and TS* = S*T, show that their sum S+T…

A: I have used just definition of normal operator

Q: Let T : V --&gt; W be linear, b ϵ W, and K = {x ϵ V: T (x) = b} be nonempty. Prove that if s ϵ K,…

A: Given: Let T : V --&gt; W be linear, b ϵ W, and K = {x ϵ V: T (x) = b} be nonempty The objective is…

Q: Prove that f, uv v • dỉ = – §̟ vũu • dĩ

A: Whenever the force is conservative in nature the work done over the closed loop is always zero. i.e…

Q: Let L:R→R be a linear operator, if L(X)= L(Y), then X=Y. Select one: O True O False

A:

Q: f || x, is Infinite norm of vector x on R then Al, = max Ax|, Select one: True False

A: We have to check whether the statement, "Given x∞ is infinite norm of vector x on Rn then…

Q: Let H be a Hilbert space, TE B(H) and define W(T) = {(r, Tr) | |r] =1}. (a) Show that o(T) C W(T).…

A: Given that: H=Hilbert space, T∈BH and WT=x,Tx  x=1. The objective is to show that, a σT⊂WT¯. b An…

Q: If V = W1 •…. Wk, then there exist k linear operators E1, … , Ek On V such that %3D (i) each E; is a…

A:

Q: a Fmiel fhe Langth and directin ofuxv and vXu by Uz } i-tj+K , V= i+j+2K

A:

Q: Show that the operators T,··, T4 from R² into R² defined by ... (§i, 52)→ (Š1,0) (§1, É2)→ (0, E2)…

A:

Q: Define T:R → R' by T(x, y, z) = (x-5y+4z, y- 6z). a) Is T one-to-one? Justify your answer. b) Is T…

A:

Q: f (て) be a top. space 3.ト X-Tab, cidiel (xて) be top space %3D %3D et A= {bj. FindO A Tb,d] %3D 2.…

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: Let v,w ɛ R" and define A = vw². Prove that ||A||2 result true for the Frobenius norm, i.e. is…

A: Let v,w∈ℝn define A=vwT. v=x1,x2,...,xn and w=y1,y2...,yn. Then,…

Q: Is every liner operator T : IR2 → IR2 diagonalizable? Justify!

A: Linear transformation and diagonalization

Q: Let V be a vector space, and let T: V →V be linear. Prove that T2 = T0 if and only if R(T) ⊆ N(T).

A: Given:Let V be a vector space and let T: V →V be linear. Prove that T2 = T0 if and only if R(T) ⊆…

Q: Let d: R2 x R2 R be defined as follows. d((x1, y1). (x2, y2)) = 3/(x1 - x2)² + (y1 - Y2)². %3D Then…

A:

Q: Let L: R → R be a linear operator such that L X2 )=| X1 then Ker(L) spanned by: Select one: а. b.…

A: Given a linear operator L from R3 to R3. We need to find the set of vectors that spans Ker(L)

Q: 9) Suppose V is a vector space and S,T E L(V,V) are such that Range S c ker T =null T. Prove that…

A: See the detailed solution below.

Q: Conversely, if T :V V is a linear operator such that T2 = T, show that V = kerT O imT. [Hint: 3.b, v…

A:

Q: -If CI(A) denotes the closure of A, which of the following is not correct in general: O (AU B)' = A'…

A: option (1) is not correct see below the reason

Q: Now Considex a fn abarce of set of finctrons f() x€ (-00gt00) such that of d del varieble Oes ox→ I…

A: As per the instruction, we will be answering only the second part. For (ii), We consider a function…

Q: (a) Verify the Cayley-Hamilton theorem. (b)  Use the theorem to express A2 and A3 as linear…

A: Part (a), Let, A=4-112 Now, the characteristic equation of A,…

Q: Let xe C and ye show that <xiy= y* AX is que inmer product if A is hermitian positive def iwite

A: Let x∈ℂm and y∈ℂm To prove that x,y=y*Ax is an inner product if A is Hermitian positive definite.…

Q: Consider the set R. Prove: (i) VxER, x 0 = 0;

A: To prove: For any x∈ℝ, x·0=0.

Q: Let L:R→ R be a linear operator such that L X2 )=x1, then Im(L) spanned by: -> X3 Select one: a. 一 O…

A: The given linear operator is: L:ℝ3→ℝ3 such that Lx1x2x3=x1x1x1

Q: Suppose L: R → R° be defined by u, +17 Lu2 2u2 %3D Uz Uz Determine whether L is a linear operator or…

A: According to question given :L:R3→R3Lu1u2u3=u1+12u2u3We have to determine whether L is a linear…