This preview shows page 1. Sign up to view the full content.
Unformatted text preview: Math 330  Homework Section 1.3 Pages 3739
(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 21 4 2 = 5 1 = 1 4 u  2v = 2 =  34 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 & R3R32R1 = 0 5 4 1 2 5 0 11 0 5 4 3 2 5 8 7 1 0 2 5 R3R3R2 = 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 ...
View
Full
Document
This note was uploaded on 09/19/2009 for the course MATH 330 taught by Professor Johnson,j during the Spring '08 term at Nevada.
 Spring '08
 Johnson,J
 Math, Linear Algebra, Algebra, Equations

Click to edit the document details