{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

quiz2-sol - augmented matrix a 1 a 2 a 3 | b is consistent...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Name: MATH227 Quiz 2 (Fall 2006) For full credit, please show all steps in details!! Let A = [ a 1 a 2 a 3 ] = 2 0 6 - 1 8 5 1 - 2 1 , b = 10 3 3 , and W be the set of all linear combinations of the columns of A (i.e., W = Span { a 1 , a 2 , a 3 } ). (a) Show that 0 and a 2 are in W . (4 points) (b) Is the vector b in W ? How about the vector a 1 + b ? (6 points) (a) Since 0 = 0 · a 1 + 0 · a 2 + 0 · a 3 and a 2 = 0 · a 1 + 1 · a 2 + 0 · a 3 , the vectors 0 and a 2 are in W = Span { a 1 , a 2 , a 3 } . (b) Consider the augmented matrix [ a 1 a 2 a 3 | b ] = 2 0 6 | 10 - 1 8 5 | 3 1 - 2 1 | 3 R 2 + 1 2 R 1 R 3 - 1 2 R 1 2 0 6 | 10 0 8 8 | 8 0 - 2 - 2 | - 2 R 3 + 1 4 R 2 2 0 6 | 10 0 8 8 | 8 0 0 0 | 0 . We see from the above row echelon form that the system of linear equations with
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: augmented matrix [ a 1 a 2 a 3 | b ] is consistent. Then, it is equivalent to say that b is a linear combination of a 1 , a 2 , a 3 . That is, b is in W . Now, as b is in W , b = c 1 a 1 + c 2 a 2 + c 3 a 3 for some scalars c 1 , c 2 , c 3 . Then a 1 + b = ( c 1 + 1) a 1 + c 2 a 2 + c 3 a 3 . Therefore, a 1 + b is also in W . 1...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern