mid 2 review

Question 1

how to determine if a set is a subspace of a vector space

Answer 1

1. zero vector in the larger space is in the set
2. set is closed under vector addition and scaler multiplication
   (or you combine for linear combination

Question 2

how to check vector addition closure

Answer 2

get 2 random vectors same size and add together

Question 3

how to check closure under scaler multiplication

Answer 3

multiply a vector by a scaler and prove that it still passes the condition needed in the set

Question 4

how to find the subset of matricies such that you get a certain output

Answer 4

1. check the zero matrix
2. check if closed under linear combonation
   (vector addition & scaler multi)

Question 5

how to determine if a set is a basis for r^3

Answer 5

1. must span r^3
2. must be linearly independent

Question 6

how to check for linear independence

Answer 6

1. make a matrix and row reduce till you see no free variable (every row has a leading 1)
2. just do the determinant if it = zero then its dependent and if it doesnt then its independed

Question 7

how to know if a vector spans a certain space

Answer 7

the rank of the matrix

Question 8

what are the 2 things that the determinate being zero proves

Answer 8

1. linear independance
2. if = 0 there is no inverse

Question 9

how to show that det(a) - plus or minus 1

Answer 9

det(A A^t) = det(I)
det(A) det(A^t)= 1
det(a) det (a) = 1

Question 10

what do you do when u get i^2 in complex numbers

Answer 10

you switch the i with (-1) bascially just changing the sign of the coefficent and removing the i variable

Question 11

what is the method of dividing 2 complex numbers

Answer 11

numerator times the conjugate of the denominator and the denomintor you take the norm squared ( basically just square the values in the denom)

Question 12

how to find the bases for row(a) in a matrix

Answer 12

see what rows have a pivot 1 ( leading 1)
then go to the orignial matrix and list out only those rows

Question 13

how to find the bases for row(a) in a matrix

Answer 13

look at the columns with leading 1 (pivots)
then go to the original matrix and list out the columns only those with the leading 1

Question 14

how to find the basis for null(a)

Answer 14

write out each row of the reduced matrix like a linear equation
solve the equation for the right row and the coeficent of the variables is the numbers we put in the matrix
( also the zero row is the freee variable t =1)

Question 15

how to find the nullity of a matrix

Answer 15

row reduced until in rref then the nullity is the number of columns that dont have a leading on (pivot)

Question 16

what is the rank nullity theorem

Answer 16

its the rank (number of leading 1) + nuility ( number of non leading 1 columns)

Question 17

is every vector in the row orthogonal to the vectors in the null

Answer 17

yes beause the dot product of vectors in row(a) and null(a) will always be 0 and always is orthogonal

Question 18

how to know if a basis is orthogonal

Answer 18

take the dot product and if it equals zero then it is

Question 19

how to find the coordinates that take 2 vectors to another vector

Answer 19

make a linear combonation of the 2 vectors with a coeficient infront of each
then make 2 equations with the numerator and the denominator and constants and then solve the constants using sub method
make a vector with the 2 numebrs found

Question 20

how to find a vector realtive to the standard basis such that ( 2 element vector)

Answer 20

you take the dot procut of the matrix u got by the vecotr they give take the 2 numbers u get from that and thats the vector your looking for

Question 21

how to find the change of basis

Answer 21

take the matrix u got then to the line down to make the right side and on the right side put in the standard matrix
then do row reduction until u get the standard on the left and the matrix on the right is the answer

Question 22

how to find a set of vectors such that ( 2x2 matrix) is the transition from w to b

Answer 22

w = b dot ( given matrix) and then take the values and make a 2x2 matrix and then spilt it into two and make 2 vectors with the columns

Question 23

how to write the standard matrix of a linear transformation and rotationg the vector pi/2 and reflecting across the x axis

Answer 23

make the refelection matrix first
( cos(pi/2) , - sin (pi/2) / sin (pi/2) , cos (pi/2))
- cos should be zero a sin 1
then to refect you switch the rows of the matrix

then to end it you take the dot product of both matrixs

Question 24

how to take the dot product of 2 2x2 matrix

Answer 24

collum of the right side matrix times the top row of first
collum 2 of the right side matrix times the top row of first
collum of the right side matrix times the bottom row of first
collum 2 of the right side matrix times the bottom row of first