Engineering Calculus Notes 43

Engineering Calculus Notes 43 - (b Now suppose −→ u...

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

View Full Document Right Arrow Icon
1.2. VECTORS AND THEIR ARITHMETIC 31 (c) Show that if a −→ v = −→ 0 then either a = 0 or −→ v = −→ 0 . ( Hint: What do you know about the relation between lengths for −→ v and a −→ v ?) (d) Show that if a vector −→ v satisfies a −→ v = b −→ v where a negationslash = b are two specific, distinct scalars, then −→ v = −→ 0 . (e) Show that vector subtraction is not associative. 4. (a) Show that if −→ v and −→ w are two linearly independent vectors in the plane, then every vector in the plane can be expressed as a linear combination of −→ v and −→ w . ( Hint: The independence assumption means they point along non-parallel lines. Given a point P in the plane, consider the parallelogram with the origin and P as opposite vertices, and with edges parallel to −→ v and −→ w . Use this to construct the linear combination.)
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: (b) Now suppose −→ u , −→ v and −→ w are three nonzero vectors in R 3 . If −→ v and −→ w are linearly independent, show that every vector lying in the plane that contains the two lines through the origin parallel to −→ v and −→ w can be expressed as a linear combination of −→ v and −→ w . Now show that if −→ u does not lie in this plane, then every vector in R 3 can be expressed as a linear combination of −→ u , −→ v and −→ w . The two statements above are summarized by saying that −→ v and −→ w ( resp . −→ u , −→ v and −→ w ) span R 2 ( resp . R 3 ). Challenge problem:...
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