chapter 4

Question 1

w is in the NulA ________________?

Answer 1

if and only if Aw=0

Question 2

V is a vector space if the following properties hold:

Answer 2

• V is closed under addition
• V is closed under scalar multiplication
• addition in V is commutative
• addition in V is associative
• additive identity (z) such that v+z=v (zero vector)
• additive inverse (w) such that v+w=z
• left and right distributive
• Iv=v

Question 3

linear combination and weights

Answer 3

c1v1 + c2v2 +….

weights => c1, c2,……

Question 4

span

Answer 4

set of all linear combinations

Question 5

subspace if:

Answer 5

• contains the zero vector
• is closed under addition
• closed under scalar multiplication

Question 6

linearly dependent vs linearly independent

Answer 6

linearly dependent if at least one of the vectors in S is a linear combination of the others vectors in s

linear independent if there is only the trivial solution

Question 7

basis

Answer 7

basis if:

• spanB=W
• B is linearly independent

Question 8

dimension

Answer 8

number of elements in the basis B of W

Question 9

linear transformation

Answer 9

• T(v+w)=T(v)+T(w)

- T(cv)=cT(v)

Question 10

kernal linear transformation T

Answer 10

kerT= {v in V | T(v)=0} subset of V

Question 11

image of the set

Answer 11

ImT = {T(v) | v in V} subset of W

Question 12

onto (surjective) linear transformation T

Answer 12

if ImT=W

Question 13

one-to-one (injective) linear transformation T

Answer 13

if T(v1)=T(v2) implies v1=v2

Question 14

T:V->W is an isomorphism

Answer 14

• T is a linear transformation
• T is one to one
• T is onto

Question 15

If T is an isomorphism then we say ______________

Answer 15

V and W are isomorphic

Question 16

skew symmetric matrices

Answer 16

satisfy the equation A + A(transpose) = [0]