Q1: Recall the definition of 1-1, norm and - norm on R². Consider the matrix A= as a linear map from R² to R². a11 912) azı azz Find a positive constant C₁ >0 such that AxSCx₂ DNE Vx ER². Conclude from here that the linear map A acting from (R²,-₁) into (R².-) is continuous.

Q: Please show me the steps to solve this question. It is a part of vector calculus.

A: Value of the integral is 9/16Explanation:

Q: Determine whether the following matrix is orthogonal or not: cosa sina A = sinßsina cosß -sinßcosa…

A: To determine if a matrix is orthogonal or not, multiply the matrix by its transpose. If the…

Q: Let A= 00 b) Say if A is diagonalisable sur Cv H E M3x3(C) such that H-¹AH is diagonal.

A:

Q: 3. Let W = M2x2(C) with zero determinant. Show that W is not a subspace of V = M2x2(C).

A:

Q: A vector y in Rn can also be viewed as an n × 1matrix Y = (y). Show that ||Y||2 = ||y||2

A:

Q: Use Axioms only to prove that the the solution to the equation 2x + y + 3z = 0 is a subspace of R³.

A: The given equation .To prove that the solution to the equation is a subspace of .

Q: 2. Let A be an m by n matrix and B be an n by n matrix. (a) If rank(A) = nullity(A), prove that n is…

A: Since you have posted multiple questions with multiple sub parts, we will provide the solution only…

Q: Let V be an m xn matrix with orthonormal columns, and x , y be in R" then prove that the linear…

A:

Q: Find the transitive closure of R={(0,1),(0,2),(1,1), (1,3),(2,2),(3,0)}

A: Given relation is                              R=(0, 1), (0, 2), (1, 1), (1, 3), (2, 2), (3, 0) we…

Q: et L be the linear operator on R? defined by L(x1, x2) = (x1 + 2x2,3x1)* Find the matrix A…

A:

Q: Do not solve just prove using grade 12 knowloge No matrix  3. Show that the planes below form a…

A: Given planes are p1:x+y–z=-2p2:2x+2y–z=6p3:4x+4y–z=-5 Let eliminating z from plane p1 and p2 we get…

Q: (1,1,..., 1)T be the column vector consisting of all 1s. Prove 0. 3. Let a = that La

A: Given: Let a=(1,1,......1)T  Then objective is to prove  La=0

Q: Find the dimension of the subspace spanned by the given vectors: u₁ = (1, 3, -2,5), u₂ = (0, 1,-1,…

A:

Q: 8. Let V={(x,y)x,y € R}. Define addition on V as follows: (x»x)+(x,y)=(x +2x, 1,y+Y,+2) Find the…

A: To find zero vector of V

Q: True/False: If A is diagonalizable, then so is AT.

A: A matrix is said to be diagonalizable if a matrix of n×n elements has n linear independent eigen…

Q: 13. Given a matrix X and a vector y of appropriate dimensions, define f(w) = ||y – Xw||² %3D a. If X…

A: given a matrix X and a vector y of appropriate dimensions, define as  fw=y-X w2 a) if X is N×k ,…

Q: 6. Let V be the vector space of all diagonal 5 x 5 matrices, then dim(V)=--

A: Given that the vector space contains all diagonal 5×5 matrices and we need to find its dimension.

Q: a) Find the value(s) of h for which the following vectors v, 0,v2 1 and vz = 2h are linearly…

A: For question (a) we will first determine the value of h for which the three vectors are not linearly…

Q: Determine the kernel and the range of the linear operator on L(x) = (x1 + x3, X2, 0)

A:

Q: Let V be a 2-dimensional vector space with two bases B and B' and let T be a linear operator on V.…

A:

Q: Is v = |1a vector in the span of S = }|2| Use Gaussian elimination to support your answer.

A: To check v=111 a vector in a span of S=123, 012, -201 or not.

Q: 19. Prove that the set of all vectors in R³that satisfy the equation x₁ = 3x2, is a subspace.

A: (.)  Let set  'S'  is given by ,        S=x1,x2,x3∈ℝ3  :  x1=3x2 or   S= x1,x2,x3∈ℝ3 :  x1-3x2=0…

Q: Let ū = -4î + 4ĵ – 3k and i = î – 25 + 2k. (a) Let w = proj, ū. Compute w. (b) Compute u-w and show…

A:

Q: a) Given the following matrices a = -and c= nd 2. 3 and C = 2. i) AT C. i) 2(BC"). i) D if (2AB)" =…

A:

Q: Consider, 21u = 2x – 3y and 21v = -–x+y Find the value of the Jacobian a (x, y) a (u, v)

A: 21 u = 2x - 3y, 21 v = -x + y

Q: d) Let A be a m × n matrix with real number entries. By considering the linear map TA : R" → R"…

A: To find - Let A be a m×n matrix with real number entries . By considering the linear map TA : ℝn →…

Q: Use matrix multiplication to find the dilation of (a, b) with factor k = α, where α > 1

A:

Q: 3. Demonstrate the following relationships involving the determinate (a) Vector cross product (b)…

A:

Q: Define the linear transformation 7: R^ → Rm by T(v) = Av. Find the dimensions of Rª and Rm. A = 4 1…

A:

Q: 3. Let A = (-1 2 and B = 2 0 I. Compute tensor products A®B and B® A: II. Compute Hadamard products…

A: Given A =01-12 & B =1220 1. we have to compute the tensor product A⊗B & B⊗A  A⊗B = 01-12⊗…

Q: (1.8) Example: Let MR) the vector space of matrices over R. for A = (a,) and b = (b,) € M (B) define…

A:

Q: True or false: If Py and Pw are the n x n orthogonal projection matrices to two orthogonal lines V,…

A: Orthogonal projection is a fundamental concept in mathematics, particularly in linear algebra and…

Q: 2. Let B₁ be the basis r²e, ze, e of V = span(r²e², xe, x), and B₂ be the basis is H.6 of w that the…

A:

Q: Let CE Rxn be a matrix such that xCx>0 for all nonzero vectors x ER". Prove that C is nonsingular.

A: