MATH 330 HW 1.3 - Math 330 Homework Section 1.3 Pages...

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Unformatted text preview: Math 330 - Homework Section 1.3 Pages 37-39 (2) Compute u + v and u - 2v u= SOLUTION: 3 2 v= 2 -1 u+v = 3 2 3 2 2 -1 + 2 -1 3 2 = 3+2 2-1 4 -2 = 5 1 = -1 4 u - 2v = -2 = - 3-4 2 - (-2) (6) Write a system of equations that is equivalent to the given vector equation: x1 -2 3 + x2 8 5 + x3 1 -6 = 0 0 SOLUTION: The system will have two equations, one for each row of the above vectors: -2x1 + 8x2 + x3 = 0 3x1 + 5x2 - 6x3 = 0 (8) Use the figure to express each of x, y, z, andw as a linear combination of u and v: 1 SOLUTION: When viewed in color, we can see the following: The red segments show that w = 2v - u The first red segment (from the origin) and the green segments show that x = 2v - 2u The first red segment (from the origin), the green segments, and the blue segment show that y = 3.5v - 2u While you cannot see the vectors in their entirety,the first red segment, the first green segment, and the purple segments show us that z = 4v - 3u. (12) Determine if b is a linear combination of a1, a2, and a3 : 1 0 2 -5 a1 = -2 , a2 = 5 , a3 = 0 , b = 11 2 5 8 -7 SOLUTION: Answering the question is equivalent to seeing whether the following system is consistent (notice we do not need to know the solution to the system): 1 0 2 -5 1 0 2 -5 R2R2+2R1 & R3R3-2R1 = 0 5 4 1 -2 5 0 11 0 5 4 3 2 5 8 -7 1 0 2 -5 R3R3-R2 = 0 5 4 1 0 0 0 2 The last row shows that this system is inconsistent. Therefore, b is not a linear combination of a1, a2, and a3. (24) TRUE or FALSE (A) Any list of 5 real numbers is a vector in R5 TRUE While we might generally insist on having a column vector as long as the order of the list is clear and we do allow the possibility that numbers be repeated, this would indicate such a column vector and hence naturally be recognized as an element of a5. (B) The vector u results when the vector u - v is added to the vector v TRUE (u - v) + v = u + (-v + v) = u + 0 = u (C) The weights c1, . . . , cp cannot all be zero in a linear combination c1 v1 + + cp vp FALSE It is isn't very interesting to have all the weights be zero since you always just get the zero vector but it is an important possibility. (D) When u and v are nonzero vectors, span{u, v} contains the line through u and the origin. TRUE This is the portion of the span which consists of linear combinations of the form u + 0v = u. 2 (E) Asking whether the linear system corresponding to an augmented matrix a1 a2 a3 b has a solution amounts to asking whether b is in Span{a1, a2 , a3} TRUE The statement x1 a1 + x2 a2 + x3a3 = b is equivalent to x1 [a1 a2 a3] x2 = b x3 This last equation is the matrix form for the system described by the original augmented matrix. 3 ...
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