PURE Flashcards
i^2 = ?
-1
i^3 = ?
-i
i^4 = ?
1
cos(-a) = ?
cos(a)
sin(-a) = ?
-sin(a)
tan(-a) = ?
-tan(a)
What is modulus argument form?
z = r(cos(a)+isin(a))
How do you invert a 3x3 matrix?
-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
If the coeffiecient matrix for a set of linear equations is singular then…
Either
-the equations have no solutions
Or
-the equations have an infinte number of solutions
If the coeffiecient matrix for a set of linear equations is non-singular then…
The equations have a unique solution
If there is a unique solution…
The three planes intersect at a single point
If there are infinitely many solutions…
Either
-The three planes intersect on a common line (sheaf)
Or
-The planes are the same
If there are no solutions…
Either
-The planes are parallel
Or
-The planes intersect to form a triangular prism
What are the 4 types of linear transformation?
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…
What does the inverse of a linear transformation do?
Reverses original transformation
What does the modulus of the determinant of the transformation matrix represent?
The area scale factor of the transformation
What is an invariant point?
A point that has been transformed onto itself
If the determinant of the transformation is negative, what does this tell you?
A reflection has taken place
What is a line of invariant points?
A line on which every point on that line is invariant
The point (x,y) is invariant if…
M(x y) = (x y)
Use this to find invariant lines
What is an invariant line?
A line which is mapped onto itself
If we are asked to find all invariant lines then we need to consider all possible lines. All possible lines are…
y = mx + c
x = k
A general point on the line y = mx + c is…
(x,mx+c)
A general point on the line x = k is…
(k,y)
If matrix M represents a 2d transformation then…
-The first column is the image of the point (1,0)
-The second column is the image of the point (0,1)
If Matrix M represents of 3d matrix then…
-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)
What is the convention of rotation?
The rotation will be clockwise or anticlockwise depending on how it appears if the axis of rotation is pointing straight at you
What are the three possible reflections?
-In x = 0 (yz-plane)
-In y = 0 (xz-plane)
-In z = 0 (xy-plane)
What is the matrix for a rotation about the x-axis?
1 0 0
0 cosA -sinA
0 sinA cosA
What is the matrix for a rotation about the y-axis?
cosA o sinA
0 1 0
-sinA 0 cosA
What is the matrix for a rotation about the z-axis?
cosA -sinA 0
sinA cosA 0
0 0 1
What is the summation formula for
1?
=n
What is the summation formula for
n?
0.5n(n+1)
What are the steps for proof by induction?
-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
What is the conclusion for proof by induction?
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
What are the steps for stronger inudction?
-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
A + B =
-b/a
AB =
c/a
A^2 + B^2 =
(A + B)^2 - 2AB
A^3 + B^3 =
(A + B)^3 - 3(AB)(A+B)
A + B + C =
-b/a
AB + AC + BC =
c/a
ABC =
-d/a
A^2 + B^2 + C^2 =
(A + B + C)^2 - 2(AB + AC + BC)
A^3 + B^3 + C^3 =
(A + B + C)^3 -3(AB + AC + BC)(A + B + C) + 3ABC
A + B + C + D =
-b/a
AB + AC + AD + BC + BD + CD =
c/a
ABC + ABD + ACD + BCD =
-d/a
ABCD =
e/a
A^2 + B^2 + C^2 + D^2 =
(A + B + C + D)^2 -2(AB + AC + AD + BC + BD + CD)
