chapter 2

Question 1

scalar product/Euclidean inner product

Answer 1

= u1v1 + u2v2

Question 2

norm

Answer 2

||v|| = ^1/2

Question 3

angles between vectors

Answer 3

=||v||·||w||cos(v,w)

Question 4

two vectors are perpendicular iff..

Answer 4

their inner product vanishes

Question 5

cauchy-schwarz

Answer 5

|<u>| ≤ ||u||×||v|| with equality iff u,v collinear</u>

Question 6

three vectors are linearly dependent if..

Answer 6

one can be written as a combination of the other two

Question 7

three vectors are linearly independent if..

Answer 7

gaussian elimination provides single solution of all coefficients = 0 (trivial)

Question 8

a linearly independent family of Rn contains _ vectors

Answer 8

a linearly independent family of Rn contains at most n vectors

Question 9

the span of a set of vectors..

Answer 9

..is the set of all of their linear combinations

Question 10

a spanning family of Rn contains _ vectors

Answer 10

a spanning family of Rn contains at least n vectors

Question 11

basis of Rn is..

Answer 11

a family of vectors such that every vector of Rn can be written uniquely as a linear combination of the family

Question 12

a family is a basis if and only if..

Answer 12

both spanning and linearly independent (a linear combination both exists and is unique)

Question 13

a basis of Rn contains _ vectors

Answer 13

a basis of Rn contains exactly n vectors