Math Study

Question 1

infinitely many solutions mean

Answer 1

there is at least one free variable

Question 2

1 solution means

Answer 2

unique solution, no free variables

Question 3

0 solutions mean

Answer 3

sys is inconsistent (a row where 0 = #)

Question 4

w a linear combination of v and u

Answer 4

[v u | w] if consistent or a free variable

Question 5

transformations

Answer 5

look at 7+8

Question 6

independent

Answer 6

pivot in every column

Question 7

columns of A span^#rowsA

Answer 7

pivot in every row

Question 8

solution to Ax=w for all w

Answer 8

pivot in every row

Question 9

Ax=0 has a unique solution?

Answer 9

pivot in every column

Question 10

one to one

Answer 10

pivot in every column

Question 11

onto

Answer 11

pivot in every row

Question 12

matrix multiplication

Answer 12

given in rxc, so if given AB, then c of A = r of B

Question 13

inverse using matrix of original and identity

Answer 13

[A|I] -> [I|A^-1]

Question 14

basis for null space

Answer 14

x such that Ax = 0, solution vector

Question 15

basis for column space

Answer 15

original independent vectors/columns

Question 16

vectors v in column space of A

Answer 16

Ax=v is consistent

Question 17

dimension of a space spanned by vectors

Answer 17

number of independent columns

Question 18

basis for space spanned by vectors

Answer 18

independent original vectors

Question 19

v in null space of matrix

Answer 19

if Av = 0

Question 20

vectors form a basis for R^m

Answer 20

linearly independent vectors (pivot in every column)
and
# of vectors = dim = m (want pivot in every row)

Question 21

vectors form a basis for H=span{v1…vk]
is w in H?
give w with respect to H

Answer 21

w = c1v1+c2v2…
[w]b = [c1 c2]

Question 22

subspace of vector space

Answer 22

think and look at 21

Question 23

verify linear transformation
kernel and range

Answer 23

think and look at 22

Question 24

polynomials and matrices independent?

Answer 24

turn into vectors
pivot in every column

Question 25

fx spanned by px and qx?

Answer 25

fx =apx+bqx, a and b exist unique solution, or many

Question 26

basis and dimension of polynomials and matrices

Answer 26

turn into vectors basis: og independent polynomial or matrix dim: # of independent vectors

Question 27

det using cofactor expansion or EF

Answer 27

det = (-1)^i+j x (aij) det(Aij) EF: swap rows (-det), row times c(1/c det)

Question 28

diagonalize

Answer 28

values: det(A-&Id) = 0 vectors: plug in values and find solutions vectors per value (nxn needs n) P are vectors and D are values

Question 29

kernel

Answer 29

vectors such that Ax = 0