2 1 0] 0 2 1 then 1 0 2 * A is Normal : A is Hermitian : A is Unitary * A is invertible. • A is diagonalizable.

Q: A*(x) = n(t, Un)un, x≤ H. Since for all x H, A* (A(x)) A(A*(x)) = Σkn(A(x), Un)Un = Σ|kn|²(x, Un)Un,…

A: Kindly refer to Step 2 for the solution.

Q: Let A be an m×n matrix with singular value de-composition U ∑ VT, and suppose that A has rank r,…

A: We will use the property of V in the SVD of matrix A to prove the claim.

Q: (9) Given that A and B are n x n matrices and (A, B) >= Tr(A" B) defines an inner product on vector…

A:

Q: Suppose that the matrix 10 1 + #1 011 x₁ + x₂ 0 0 0 1+ ₂+ 3 the result of reducing the matrix [u v w…

A: Given reduced matrix is  u v w  =101011000 Since we get two pivots (marked in red ) after reducing…

Q: (b) Let B = {₁,...,n} be a basis for an inner product space V with inner product (,). We can nxn…

A: It is given that B=b1, b2, ⋯, bn is a basis for the inner product space V with inner product .. The…

Q: (:) X1 + 2x2 2x1 + x2 X1 + x2 Let L: R? → R³ be defined by L Let B1 and B2 be ordered bases for R?…

A: Remark: Let L : Rn -> Rm be a linear transformation, then the matrix representation of L will be…

Q: For every real m x n matriz mx m) and a diagonal m x n matriz

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Q: Which of the following is a basis for the subspace {:)- x y z W = x = y + z and x, y,z E R}C M2x3(R)…

A: We have to satisfy the given condition in the matrix i.e. x = y+z

Q: suppose there is a matrix A, A= 60 0 -3 O оо 6 30 03 о 0 4x4 1. proof there are projections V₁, V₂,…

A:

Q: 6: Let V be the vector space of all symmetricnx n matrices over R. Then dimension of V is n(n - 1)…

A: Please refer to the below-uploaded images to step by step solution to your problem.

Q: find the Linear trans fromation Tee whose matrix Wzxz i -1 Hi- + =1421] -2 with regard to the base H…

A: Question had typo. Hence solved after correction

Q: 5. For which value of x do the following vectors form a linearly dependent set in ² (글 글) (글 글(글 글)

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Q: 1.2.7 The dimension of a vector space depends on its ground field. What is the dimension of C" as a…

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Q: D: P2> P2 (Dthe differential operator so that D(ax²+bx+c) = 2ax +b); a the standard basis for P2; B…

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Q: Let T: R³ → R², C canonical base of R² and B = basis of R³. • ( 1 ) = (x + 2), ² (y t then [7]…

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Q: We define a linear bijection, h, between R4 and H(2), the set of complex 2 x 2 hermitian matrices,…

A:

Q: Suppose that x and y are mxn matrices. Prove that xT- yT=(X -YT. Find a 2x2matrix A, that is not the…

A: Take A and B as arbitrary m×n matrices and evaluate the left and right sided expression

Q: Let A=[aij ] be an m xn matrix and k be a scalar then kA is equal to : * [kaij]mxn [aij/k]mxn…

A:

Q: Suppose TEL(V) has singular value decomposition given by Tv = $₁(v. e₁) f₁ +...+ Sn(v.en) fn for…

A: The problem is solved using the concept of singular value decomposition.

Q: 79. Agriculture A fruit grower raises two crops, apples and peaches. The grower ships each of these…

A:

Q: not and en- -0 37. Let A = 2 3 5 Construct a 2 x 3 matrix C (by trial and error) using only 1,-1,…

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Q: Let T : R? → R´be defined by [2 _5||X1 X2 1 6] [x2] а If B = is a basis for R“ such that |T'|B is a…

A:

Q: tet A be an mxn matrix Prove that on

A: Let A be an m×n matrix prove that Frobenis norm : AF=∑i=1m∑j=1naij212↔·2 on Cmn

Q: Consider S:RR" D show using det, that s is L.7. linear transtermation Determine malcix [s],

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Q: If A is an n x n matrix such that A = - A' and all of the eigenvalues of A are 0 or 1, then A is the…

A:

Q: Let Z be the linear operator on R² defined by L( i). Find the matrix A representing L with respect…

A: Basis: A set of vectors which is linearly independent and spans the whole vector space is calledthe…

Q: Let A be a square matrix, and that k1, k2 and A are scalars. Show that if Av = Xv and Aw = Xw then,…

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Q: Q1.One of the following is correct (a) are basis for the S(R’) (b))| are basis for the S(R²) and and…

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Q: Let I be the linear operator on R³ defined by L a Find the matrix A representing L with respect to…

A: As linear transformation is hence with the cofficients we get the matrix,

Q: (i) Find the change-of-basis matrix Ps,-E and then use it to find the component vector [C]s, of -…

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Q: In the vector space P of polynomials of degree atmost two, the third column of the matrix of…

A: The solution is given below in the next step:

Q: Let A be an m x n matrix with orthonormal columns, and x, y be in R" then prove tat the linear…

A:

Q: W= {x: where x are eigenvectors of an A } Is W a subspace of R" ? Show the prove

A: See solution below

Q: Linear Alegbra and Differential Equations

A: SOLUTION 5 :To find the matrix [T]σσ​ for the linear transformation (T) with respect to the standard…

Q: For each of the following matrices A E M,(R), test A for diagonalisabil- ity. If A is…

A:

Q: 1/2 If A? is a zero matrix, then AT A = 0 is: O a. Not possible unless A is a zero matrix Ob. not…

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Q: Find the smallest operator T: R³ R³ integer value for K so that the linear 3 given by T(x,y,z) =…

A: Consider the given 3D linear transformation, T:ℝ3→ℝ3, Tx,y,z=x+y+z,kx+y-z,x+y+z. Now if the 3×3…