To avoid any confusion, unless specified otherwise, vector spaces are complex vector spaces, inner products are complex inner-products, and matrices are complex matrices. True or False: A is unitary iff AH = A−1 .

Q: Show that if B is an n × n matrix with entries in C and  B3=B then B is diagonalisable. Where C is…

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Q: If A and B are arbitrary real m X n matrices, then the mapping (A, B) = trace(A" B) defines an inner…

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Q: Consider the set of all 3x3 matrices. Prove that with standard addition and standard multiplication…

A: The given question is solved below.

Q: If A is diagonalizable and for all eigenvalues, A of A, |A| = 1, then A is unitary.

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Q: Let S be the set of matrices of the form a -a :). -b b where a and be R. The vector addition and…

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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…

Q: 1.) Let V be the set of all 2x2 matrices A = [ such that the product abcd = 0,1 as the operation O…

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Q: Exhibit a basis for T2, the vector space of all 2 x 2 matrices of real numbers with zero trace.

A: Let V be a vector space over a field F. A set S of vectors in V is said to be a basis of V if  (i) S…

Q: Prove that the inverse of an orthogonal matrix is orthogonal.

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Q: Which of the following sets are closed under scalar multiplication? (i) The set of all matrices in…

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Q: Consider the matrices, 1 i V1 – 6² 0 -6 A = 1 а 1 B = 1 1 0 V1– 62 a where a and b are real…

A: Given, the matrices                              A=a1i1a1-i1a and B=1-b20-b010b01-b2 Then, we have…

Q: Consider the following block matrix: N a aT c where N is a m ×m real symmetric matrix, a is a…

A: We will use the basic knowledge of matrices to answer this question.

Q: Let A and C be n×n matrices such that ​CA=I​ (the n×n identity​ matrix). Show that Ax​=0 has only…

A: Here, CA=I, where I is the identity matrix.

Q: Let M be the vector space of 2 x 2 real-valued matrices. For M = [ a b c d] we define M# = [d b…

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Q: Which of the following is not a vector space? (; :) A. The set of all 2x2 matrices of the form with…

A: Solution:-

Q: Prove that the matrices and a a a 0. a 3. a are Similar ( These 2 matrices are not deagonalısable)

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Q: If v is an eigenvector of A, then ū is an eigenvector of A".

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.