Section 6 Notes - Suggested problems (Chapters 4.1-5.2):...

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MATH 294 Chapter 4.1 Introduction to Linear Spaces Ex 4.1.31 Find a basis of the space of all 2 × 2 matrices S such that " 0 1 1 0 # S = S " 1 0 0 - 1 # (1) Solution: Let us write matrix S as S = " s 1 s 2 s 3 s 4 # . Then (1) can rewritten as " s 3 s 4 s 1 s 2 # = " s 1 - s 2 s 3 - s 4 # . Two matrices are equal if and only if their components are equal. That is, we get s 3 = s 1 s 4 = - s 2 s 1 = s 3 s 2 = - s 4 Taking thin into account, we get S = " s 1 s 2 s 1 - s 2 # = s 1 " 1 0 1 0 # + s 2 " 0 1 0 - 1 # for any s 1 , s 2 R . In other words, a basis is { " 1 0 1 0 # , " 0 1 0 - 1 # } . =================================================================
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Unformatted text preview: Suggested problems (Chapters 4.1-5.2): 4.1: 3,7,10,20,26 4.2: 5,10,12,28,40 4.3: 1,2,7,29,34,41,55,57,60 Feel free to solve more;). 5.1: 11,17,21,28,31 5.2: 4,20,32,38,39,40 Have a nice weekend ! ================================================================= 1...
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