{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

WI08Notes-Jan16-lincomb

# WI08Notes-Jan16-lincomb - v k ∈ U The aﬃne subspace...

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

Math 1b — Types of linear combinations A vector u is a linear combination of vectors v 1 , v 2 , . . . , v k when there are scalars c 1 , c 2 , . . . , c k so that u = c 1 v 1 + c 2 v 2 + . . . + c k v k . A nonempty set U of vectors is a subspace (or linear subspace ) when U contains all linear combinations of vectors v 1 , v 2 , . . . , v k U . The span of v 1 , v 2 , . . . , v k is the set of all linear combinations of v 1 , v 2 , . . . , v k . A vector u is a nonnegative linear combination of vectors v 1 , v 2 , . . . , v k when there are nonnegative scalars c 1 , c 2 , . . . , c k so that u = c 1 v 1 + c 2 v 2 + . . . + c k v k . A nonempty set U of vectors is a convex cone when U contains all nonegative linear combinations of vectors v 1 , v 2 , . . . , v k U . The convex cone generated by v 1 , v 2 , . . . , v k is the set of all nonnegative linear combinations of v 1 , v 2 , . . . , v k . A vector u is an aﬃne combination of vectors v 1 , v 2 , . . . , v k when there are scalars c 1 , c 2 , . . . , c k so that c 1 + c 2 + . . . + c k = 1 and u = c 1 v 1 + c 2 v 2 + . . . + c k v k . A nonempty set U of vectors is an aﬃne subspace when U contains all aﬃne combinations of vectors v 1 , v 2 , . . . ,
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: v k ∈ U . The aﬃne subspace generated by v 1 , v 2 , . . . , v k is the set of all aﬃne combinations of v 1 , v 2 , . . . , v k . A vector u is an convex combination of vectors v 1 , v 2 , . . . , v k when there are nonneg-ative scalars c 1 , c 2 , . . . , c k so that c 1 + c 2 + . . . + c k = 1 and u = c 1 v 1 + c 2 v 2 + . . . + c k v k . The convex hull of v 1 , v 2 , . . . , v k is the set of all convex combinations of v 1 , v 2 , . . . , v k . A nonempty set U of vectors is an convex set when U contains all convex combinations of vectrors v 1 , v 2 , . . . , v k ∈ U . The parallelopiped spanned by v 1 , v 2 , . . . , v k is the set of all linear combinations u = c 1 v 1 + c 2 v 2 + . . . + c k v k where 0 ≤ c i ≤ 1 for all i = 1 , 2 , . . . , k ....
View Full Document

{[ snackBarMessage ]}

### What students are saying

• 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.

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

• 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.

Dana University of Pennsylvania ‘17, Course Hero Intern

• 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.

Jill Tulane University ‘16, Course Hero Intern