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Solutions to Problem Sets 1 Three Questions at the End of the Preface Gilbert Strang, Introduction to Linear Algebra, 6th Edition (2023) 1. When can lines of lengths r,s,t form a triangle? They must satisfy the strict triangle inequalities r < s+t s < r +t t < r +s If we allow equality, the triangle will have angles of 0,0 and 180 degrees.
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1 The combinations give (a) a line inR 3 (b) a plane inR 3 (c) all ofR 3 . 2 v+w= (2,3)andv−w= (6,−1)will be the diagonals of the parallelogram with vandwas two sides going out from(0,0). 3 This problem gives the diagonalsv+wandv−wof the parallelogram and asks for the sides: The opposite of Problem 2. In this examplev= (3,3)andw= (2,−2). 43 v+w= (7...
1 u·v=− 2 .4 + 2 = 0,u·w=−.6 + 1 = 1,u·(v+w) =u·v+u·w= 0 + 1,w·v= 4 + 6 = 10 =v·w. 2 ‖u‖= 1and‖v‖= 5and‖w‖=
13 s 1 + 4s 2 + 5s 3 = (3, 7 ,12). The same vectorbcomes fromStimesx= (3, 4 ,5): 7 All three rows are perpendicular to the solutionx(the three equationsr 1 ·x= 0and r 2 ·x= 0andr 3 ·x= 0tell us this). Then the whole plane of the rows is perpendicular tox(the plane is also perpendicular to all multiplescx). 8 x 1 −0 =b 1 x 2 −x 1 =b 2 x 3 −x 2 =b 3 ...
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Step-by-step solution. Step 1 of 5. The objective is to describe geometrically (line, plane or all of) all linear combinations of the following given vectors: (a) and. (b) and. (c) , and. Step 2 of 5. We know that the linear combination of two vectors v and w is given by, where c and d are two scalars. (a) Let us consider the following vectors;
Exercise 10. Exercise 11. Exercise 12. At Quizlet, we’re giving you the tools you need to take on any subject without having to carry around solutions manuals or printing out PDFs! Now, with expert-verified solutions from Introduction to Linear Algebra 5th Edition, you’ll learn how to solve your toughest homework problems.
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