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

Q: A(x + y) = Ax + Ay. 2. A(ax) = a(Ax) = (aA)x

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A: Given: ⟨u, v⟩ = (Au)·̇(Av) for u, v ∈Rn and A is an invertible n ×n matrix. To show if <u, v>…

Q: Let {v1, v2, ..., vn} be an orthogonal set in R" with respect to the Euclidean inner product. Let A…

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Q: Consider the 2 X 2 matrices A1 =  1      1           A2 =  0      1        A3  =   0      0      A4…

A: Thus, the four matrices are not linearly independent.

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Q: Let fi = 1+a + 5a², f2 = 1 + 2x + 8a². B = (f1, f2) is a basis for the subspace V = {ao +a;# + azw²…

A: s = - 10 (option a) The detailed solution is as follows below:

Q: (2) Let M₂×2 be the vector space of all 2X2 matrices, and defined T:M 2x2 (A)=A+AT. where A= a [cd]…

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A: Linear operator should satisfy L:Rn→Rn  a) Lu+v=Lu+Lv         u,v∈Rn b)Lcu)=cL(u

Q: Show M={A in M2,2| |A| =0} is not a vector space under the usual addition and scalar multiplication…

A: We will show M={A in M2,2| |A| =0} Is not vector space by exhibiting two elements A,B belongs M but…

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A: Introduction A square matrix is said to be an orthogonal matrix if…

Q: Please provide detailed solution for II part asap to get a upvote

A: Given - Let A, B ∈ Mn,nℝ be two square matrices. To Prove - Prove that if the matrices A and B are…

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: Let A be an m × ʼn matrix with real number entries. Let N(A) be the null space of A, and let Col (A)…

A: The null space of a matrix A is represented as  where V is thevector space.The transpose of a matrix…

Q: Prove that an m X n matrix A has rank 1 if and only if A can be written as the outer product uvT of…

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Q: 4. Let A be an m x n matrix and b a non-zero m × 1 matrix. Show that the set of all n x 1 (column)…

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Q: 2. Let M2x2 (R) be the set of all 2 x 2 matrices (with real number entries). That is a b {[%] | a,…

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Q: Let ¹ = [ 8 ], x= [ ¦ ! ] x = [ 9 ] -[69]-[88]-[8][9] Show that these four matrices span M2(C), the…

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Q: The equatiokn  Y = (x,y)T  is given as a vectorfunction with 2 components and A = table below 3 2 2…

A: Given is a matrix A and we have to diagonalize the matrix A. First we find eigenvalue of matrix A…

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Q: 8. = - Prove that the set is a basis for the vector space of 2x2 matrices

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Q: d) If A and B are 4 x 4 nonsingular matrices such that |2A¬'B²| = 32, then |4| = 8 e) Let S =…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

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A: Please check the next step for details

Q: 20) Let W be the vector space of all symmetric 2 x 2 matrices. Show that W sisomorphic to R³

A: Introduction: The set of vectors v1, v2, ..., vn in V is said to be linearly independent if the only…

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A: Given: If u and v are orthonormal vectors To Show: P = uuT + v vT is the standard matrix of an…

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A: Since u is orthogonal to each row of A, it means u is in null space of A.

Q: Let A be a 9x6 matrix with nullify equal to 1 a) find the rank and nullify of A^T  b) suppose that…

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Q: Q4 let V=M₂x₂ of the form [a of with standard matrix addition and multiplication form -space-

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Q: 3 Let M₂(R) be the vector space of all 2 × 2 matrices with entries from R. Show that M₂ (R) = W₁ W₂,…

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Q: Use the Wronskian to determine if the function vectors 7,= x? + e* and 72= e²x are linearly…

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Q: Show that there do not exist scalars c₁, c₂, and C3 such that c₁(1,0, 1,0) + c2 (1, 0, 2, 1) +…

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Q: Suppose V={v1,v2,...,vn} is a set of nonzero vectors, and let                    x= a1v1 + a2v2…

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Q: Let x = (−3, 1, 7, 2, 3, − 1)T, find P, Householder matrix, and a scalar a, such that Px = y, where…

A: A Householder matrix is a type of orthogonal matrix that is used to perform Householder…

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

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