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exercises that can be traced back, in one form or another, through generations of linear algebra texts, making any serious attempt at proper attribution quite futile. If anyone feels slighted, please
1. Math 262 Exercises and Solutions (1) Let A be a 3 × 3 matrix with real entries. Prove that if A is not similar over R to a triangular matrix then A is similar over C to a diagonal matrix. Proof. Since A is a 3 × 3 matrix with real entries, the characteristic polynomial, f(x), of A is a polynomial of degree 3 with real coefficients. We know ...
Problem Sets with Solutions. MIT OpenCourseWare is a web based publication of virtually all MIT course content. OCW is open and available to the world and is a permanent MIT activity.
Do this in (ii) and (iii) by expressing the subspace as the null space of an appropriate matrix, and nding a basis for that null space by nding the reduced echelon form. In each case, state the dimension of this subspace. All vectors whose entries are all the same. All vectors whose entries add to zero. 2 x1 3.
Linear Algebra is a text for a first US undergraduate Linear Algebra course. You can use it as a main text, as a supplement, or for independent study. It is Free. The book, the complete answers to all exercises, classroom presentation slides, and a lab manual using Sage, are all available for download.
Linear Algebra - Exercises 9 Linear Systems. Method of Elimination 1. Determine whether a) x =1, y = −1 b) x =2, y = −3 are solutions to the linear equation 2x+3y = −1. Solution. a) 2·1+3·(−1) = −1, −1=−1, so x =1, y = −1 is a solution. b) 2·2+3·(−3) = −1, −5 6= −1, so x =2, y = −3 is not a solution. 2.Determine if ...
Solutions to Exercises 3 17 The point 3 4 v + 1 4 w is three-fourths of the way to v starting from w. The vector 1 4 v + 1 4 w is halfway to u = 1 2 v + 1 2 w. The vectorv +w is 2u (the far corner of the parallelogram). 18 The combinations cv +dw with 0 ≤c ≤1 and 0 ≤d ≤1 fill the parallelogramwith sides v and w.
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