old mt

Question 1

how to find the determinate of a 2x2 matrix

Answer 1

top left multiplied by bottom right - bottom left times top right

Question 2

how to find the determinate of a 3x3 matrix

Answer 2

start at the top left then go up then add them together + start at tope left go up and subtract each other

Question 3

what does det = 0 prove

Answer 3

that the matrix is linearly dependent

Question 4

what if the det is not equal to zero

Answer 4

then the matrix is linearlly independent

Question 5

properties p^-1 AP = B show det(a) = det(b)

Answer 5

write the equation take the determinate of both sides expand then factor then use the fact that p^-1 = 1/det(p) to cancel the p out in the equation and then ur left with det(b) = det(a)

Question 6

what to keep in mind when multiplying complex numbers

Answer 6

i^2 you take away the i leaving just the coefficent but change the sign of the coefficent

Question 7

(1-i)^5

Answer 7

first take the norm then put into theta = arctan(b/a) blah blah idk

Question 8

how to find the cube roots (8)

Answer 8

solve for the 3 cases of k - 0,1,2
1. the regular cube root of the number - 2
2. theta 0+2pi x k /3 ( needed for next equation)
3. 2( cos(theta) + i sin(theta)

repeat for all three cases of k

Question 9

rank nullity theorem

Answer 9

rank of matrix + dimension of null ( most of the time dim = 1)

Question 10

are the vectors in the null always orthogonal to the vectors in the basis of the row(a)

Answer 10

yes it always is

Question 11

what is orthogonal

Answer 11

when the dot product is zero

Question 12

what is orthanormal

Answer 12

when its orthogonal and the norm - sqaure root(a^2+b^2) = 1

Question 13

how to find the coordinates of a vectore relative to a standard basis

Answer 13

make a matrix with the vectors give at the begging then put the standard basis beside it then do row reduction till you het the standard basis on the left and whats on the right is now the answer. take the dot product of this matrix and the vector that was given in this question and the 1x2 vector u get is the answer.

Question 14

find a vector relative to the standard basis

Answer 14

just take the dot product of the matrix thats given at the begging by the vector given in the question

Question 15

how to find the cahnge of a basis matrix from standard to the basis

Answer 15

do the same where you put the matrix beside the standard matrix and rowreduce till u get the standard then you factor out so that you get all ones

Question 16

how to divide a complex number

Answer 16

put in a fraction then multiple top and bottom by the conjugate