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FM > PURE > Flashcards

PURE Flashcards

1
Q

i^2 = ?

A

-1

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2
Q

i^3 = ?

A

-i

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3
Q

i^4 = ?

A

1

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4
Q

cos(-a) = ?

A

cos(a)

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5
Q

sin(-a) = ?

A

-sin(a)

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6
Q

tan(-a) = ?

A

-tan(a)

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7
Q

What is modulus argument form?

A

z = r(cos(a)+isin(a))

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8
Q

How do you invert a 3x3 matrix?

A

-Find the determinant
-Replace every element in the matrix by its minor
-Switch the sign of alternate elements
-Transpose the matrix
-Divide by the determinant

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9
Q

If the coeffiecient matrix for a set of linear equations is singular then…

A

Either
-the equations have no solutions
Or
-the equations have an infinte number of solutions

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10
Q

If the coeffiecient matrix for a set of linear equations is non-singular then…

A

The equations have a unique solution

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11
Q

If there is a unique solution…

A

The three planes intersect at a single point

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12
Q

If there are infinitely many solutions…

A

Either
-The three planes intersect on a common line (sheaf)
Or
-The planes are the same

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13
Q

If there are no solutions…

A

Either
-The planes are parallel
Or
-The planes intersect to form a triangular prism

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14
Q

What are the 4 types of linear transformation?

A

REFLECTION in the line…
ROTATION about the origin by … anti/clockwise
STRETCH parallel to the x/y axis by scale factor…
ENLARGEMENT with centre (0.0), by scale factor…

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15
Q

What does the inverse of a linear transformation do?

A

Reverses original transformation

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15
Q

What does the modulus of the determinant of the transformation matrix represent?

A

The area scale factor of the transformation

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15
Q

What is an invariant point?

A

A point that has been transformed onto itself

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15
Q

If the determinant of the transformation is negative, what does this tell you?

A

A reflection has taken place

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16
Q

What is a line of invariant points?

A

A line on which every point on that line is invariant

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17
Q

The point (x,y) is invariant if…

A

M(x y) = (x y)
Use this to find invariant lines

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18
Q

What is an invariant line?

A

A line which is mapped onto itself

19
Q

If we are asked to find all invariant lines then we need to consider all possible lines. All possible lines are…

A

y = mx + c
x = k

20
Q

A general point on the line y = mx + c is…

A

(x,mx+c)

21
Q

A general point on the line x = k is…

A

(k,y)

22
Q

If matrix M represents a 2d transformation then…

A

-The first column is the image of the point (1,0)
-The second column is the image of the point (0,1)

23
Q

If Matrix M represents of 3d matrix then…

A

-The first column is the image of the point (1,0,0)
-The second column is the image of the point (0,1,0)
-The third column is the image of the point (0,0,1)

24
Q

What is the convention of rotation?

A

The rotation will be clockwise or anticlockwise depending on how it appears if the axis of rotation is pointing straight at you

25
Q

What are the three possible reflections?

A

-In x = 0 (yz-plane)
-In y = 0 (xz-plane)
-In z = 0 (xy-plane)

26
Q

What is the matrix for a rotation about the x-axis?

A

1 0 0
0 cosA -sinA
0 sinA cosA

27
Q

What is the matrix for a rotation about the y-axis?

A

cosA o sinA
0 1 0
-sinA 0 cosA

28
Q

What is the matrix for a rotation about the z-axis?

A

cosA -sinA 0
sinA cosA 0
0 0 1

29
Q

What is the summation formula for
1?

A

=n

30
Q

What is the summation formula for
n?

A

0.5n(n+1)

31
Q

What are the steps for proof by induction?

A

-Check the result is true for the base case
-Assume result is true when n = k
-Use this assumption to show that the result is true for n = k + 1
-Conclusion

32
Q

What is the conclusion for proof by induction?

A

We have shown that if the result is true for n=1 then it is true for n = k + 1
As the result is true for n = 1, it is therefore true for all positive integers

33
Q

What are the steps for stronger inudction?

A

-Check two base cases
-Assume it is true for n = k and n = k + 1
-Show the result is true for n = k + 2
-Tweaked conclusion

34
Q

A + B =

A

-b/a

35
Q

AB =

A

c/a

36
Q

A^2 + B^2 =

A

(A + B)^2 - 2AB

37
Q

A^3 + B^3 =

A

(A + B)^3 - 3(AB)(A+B)

38
Q

A + B + C =

A

-b/a

39
Q

AB + AC + BC =

A

c/a

40
Q

ABC =

A

-d/a

41
Q

A^2 + B^2 + C^2 =

A

(A + B + C)^2 - 2(AB + AC + BC)

42
Q

A^3 + B^3 + C^3 =

A

(A + B + C)^3 -3(AB + AC + BC)(A + B + C) + 3ABC

43
Q

A + B + C + D =

A

-b/a

44
Q

AB + AC + AD + BC + BD + CD =

A

c/a

45
Q

ABC + ABD + ACD + BCD =

A

-d/a

46
Q

ABCD =

A

e/a

47
Q

A^2 + B^2 + C^2 + D^2 =

A

(A + B + C + D)^2 -2(AB + AC + AD + BC + BD + CD)

FM (2 decks)

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