Nullspace (Kernel), Image and ransformS 1. For the matrix A, let T: R* > R' be defined by T(x) = Ax for each in R. A is given: 31

Q: %3D 人 3. 2. ☺ Find an invertible matrix P such that D = p'Mp. Evaluate p and check if pMp -イ…

A: First we need to find the eigen  values λ1,λ2 and corresponding  eigen vectors v1,v2 respectively…

Q: Assume that T is a linear trans formation. Find the standard matrix of T for T : R² → R² first…

A:

Q: For A a square matrix such that ATA = I show that det A = ±1.   If A is square and A5 = 0 show…

A: Given, ATA=IdetATA=detIdetATdetA=1  Since, detI=1detA·detA=1    Since, detAT=AdetA2=1detA=±1 This…

Q: 2. Prove that: k(4+ mB) - kA+(km) B for any mx n matrices A and B, and scalars k and m. (You may not…

A: 2 We have to prove that kA+mB=kA+kmB Where, A and B are m×n matrices and k and m are the scalars.…

Q: Find the standard matrix for the reflection T of R'in r= 21 the line {y=-t. Your work must be shown!…

A: The solution is below.

Q: Prove that the function f(x) = defined on the domain by the condition a= [0,1], is a " α concave…

A: We will use the basic knowledge of sets and functions to answer this question correctly and…

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: 5. Find the three by three matrix A such that [r – y] A. y for all r, y, z in R.

A: In the given problem we have to find the three by three matrix A such that…

Q: Find the outer product form of the SVD for the given matrix. (Enter sqrt(n) for vn.) 5 0 A = 0 5 0

A:

Q: 10. Let A be an n x n matrix such that for every x ER", ||Ax|| = |x|| (the norm here is the one…

A:

Q: Let A denote the 2 x 2 matrix 2 We define a function (•, ·) : R² × R² → R by taking 1 (x, у) %— уТ…

A: We have A=1221. We are given the function ·, · : ℝ2×ℝ2→ℝ2 by taking x, y=yTAx for each pair of x,…

Q: : Write out the Hessian matrix at all critical points wvhich are saddle points for the function f(x,…

A:

Q: (E)-L - 2x1 Let T : IR → R² be defined by T X2 -3x2 + x3 -6° 5. Let B = { u1 ,U2 u3 and C = V2 What…

A: Let T: R3 - R2 be defined by T and we find argument matrix. And the matrix representation of T with…

Q: Let A be an n xna diagonalizable matrix. Prove that the matrix exponential satisfies det(e^) =…

A: According to the given information, Let A be an n × n diagonalizable matrix. It is required to prove…

Q: 4. Let A be an n x n matrix such that A31 = 0n. (a) Show that In - A is invertible. (b) Let B be an…

A:

Q: Prove that the function f(x)=Lnx, defined on the domain is a concave function. For this purpose,…

A: Follow the steps.

Q: Let the matrix A of size N x N, given by A = kl, with I be the identity matrix of size Nx N and k a…

A: note :  As per our company guidelines we are supposed to answer ?️only first 3 sub-parts. Kindly…

Q: "rove that function f(x, y) = is a convex function. For this purpose, compute the Hessian matrix of…

A:

Q: (E)-) f------{} 4x1 – x2 Let T : R³ → R² be defined by T 12 -2x3 Let B = { u1 7,u2 -3,u3 4 and C =…

A: Given T:ℝ3→ℝ2 be a linear transformation defined by  Tx1x2x3=4x1-x2-2x3              (i) Let…

Q: 4. Let f1, f2, fz be functions in F(R). For a set of real numbers x1, x2, X3, let (f(x,)) be the…

A: Note:- Here, you don't mention field TR is ℝ or anything else. So, i'm assuming this ℝ. If it is…

Q: (b) Let A be an n x n matrix. Show that for any positive integer t we have t-1 ( 1-A)ΣΑ* =I -Α' k=0

A:

Q: f'(0) f'(1) – f(1)/ 1. Let T : P2(R) → M2x2(R) be defined by T(f) = ( (a) [10 pts] Find the null…

A:

Q: (b) Evaluate f. F•dr by using Stoke's Theorem, where F = zi+ xj + yk and C is perimeter of closed…

A:

Q: Let A = (a;;)ij be an m x n matrix of real numbers, and let f : R" -→ Rm be defined by x2 X2 f = A…

A:

Q: Use Green's Theorem to evaluate S. axy dx + cxy dy, where C is a counterclockwise oriented…

A:

Q: matrices over K. For each j, 1 <j<n, the function E defined by

A:

Q: 4. Let A be a 3 x 4 matrix and B be a 4 x 5 matrix such that ABx = 0 for all x E R°. a. Show that…

A:

Q: 1- nullspace af now equivalent to the for the 1 1- 2. C. that puy -2 B. assume matrix pand Let A

A:

Q: Let a be a real number. Let 1 1 A = | 0 1 1 1 1 2 a) Find the value of a for which the matrix is not…

A:

Q: Diagonalize the quadratic forms by finding an orthogonal matrix Q such that th e change of variable…

A: Given: The quadratic form is given as  f(x, y) = x2 + 8xy + y2

Q: 2. Find a real 2 × 2 matrix A (other than A =I) such that Aº = I . %3D

A:

Q: 2 Let B = then B is invertible and the function S : M2(R) → M2(IR) 8. defined by S(A) = BAB-1 is a…

A:

Q: Let A be a 2 × 2 matrix, and let LA be the linear operator defined by LA(x) = Ax Show that (a) LA…

A: Both parts can be established by direct calculations for 2x2 case. The same results are true in nxn…

Q: xa - 1 Prove that the function f(x) = α concave function. For this purpose, compute the Hessian…

A: Please check the answer innext step

Q: [1 1 1 x is invertible. %3D Determine the values of x so that the matrix A

A: To determine the value of x so that the matrix A=11x1xxxxx is invertible. Solution: A matrix is…

Q: (c) Let A be an n x n matrix. Show that if (I – A) is invertible and limț∞ Aª = 0 then (I – A)-! =A*…

A:

Q: If A is n x n invertible matrix the O a. Dimension of N(A)=Dn Ob. Dimension of N(A) -Dimension of…

A: From given statement

Q: | 4B 3B] Consider L 4p ] Find all the values of the parameter ß for which the matrix is invertible.

A:

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…

Q: 7. Let A be an invertible n xn matrix over R. Consider the system of linear equation Ax = b or…

A:

Q: Let A be a 6 x 4 matrix and W= col(A). II f dim(W) = 4 what is rank(AT)? %3D (Hint: Use Theorem…

A:

Q: atrix L nas the lemen R2 – R1) → R2, R2 + R1 pplied to it to transform it to the matrix

A:

Q: For singular value decomposition of a complex matrix (M: Usigma V^T) , how do you find the U, sigma,…

A: The solution is in step 2

Q: Gve an example of a dx2 matrix A and 2-dimensional vectors u and v such that u and v are orthogonal…

A: We have to give an example of a 2×2 matrix.

Q: :Prove that function f(x, y) =: is a convex function. For this purpose, compute the Hessian matrix…

A:

Q: : Prove that the function f(x)=Ln x, defined on the domain R₁, is a concave function. For this…

A: Here Hessian matrix going to be 1×1 matrix.

Q: Let A be an n xn matrix and f(r) = Cr* + ... + cr + co, a polynomial. By f(A) we mean the n x n…

A: Please see below picture for detailed solution.

Q: Complete the matrix A so that it is Hermitian. 1 i 7 – 3i A = -3 1 4 Y = Z =

A:

Q: Linear algebra

A: To transform the given matrix into Jordan canonical form