Chapter 5 - Matrices 1 Flashcards
Matrix Properties, Arithmetic, Transformations, and Equations... what's it going to be? The red pill? Or the blue pill?
Describe a matrix of an order n x m.
How do determine the order of a matrix?
A matrix with n rows and m columns has order n x m.
What is possible on two matrices that have the same order?
If two matrices have the same order then they can be added or subtracted by adding or subtracting their corresponding elements.
What is the rule regarding matrices being multiplied by constants?
To multiply a matrix by a constant, multiply each of its elements by that constant.
What is the zero matrix?
The zero matrix is a matrix of any order with all elements equal to zero.
If two matrices can be multiplied, what are those matrices known to be able to do?
If two matrices can be multiplied, then it is said that they are conformable for multiplication.
What is the order of the product of matrices n x m, and m x p?
The product of matrices n x m and m x p has order n x p.
What is the rule for squaring matrices?
It is only possible to square a matrix if it is a square matrix, meaning the matrix has the same number of orders as rows.
How do you transpose a matrix?
You transpose a matrix by swapping the rows and columns.
How do you find the image of a vector?
In order to find the image of a vector (x,y) under a transformation T = (a b, c d) (2x2), we premultiply the vector by the matrix. (T x vector)
How do you represent a reflection in the line y = x?
The matrix
( 0 1 )
( 1 0 )
represents a reflection in the line y = -x.
How do you represent a reflection in the line y = -x?
The matrix
( 0 -1 )
( -1 0 )
represents a reflection in the line y = -x.
What is the identity matrix?
The matrix, I =
( 1 0 )
( 0 1 )
is the identity matrix, which works on all 2x2 matrices. Multiplying this transformation has no effect on the original vector.
What matrix represents a stretch of scale factor k parallel to the x-axis?
The matrix
( k 0 )
( 0 1 )
represents a stretch of scale factor k parallel to the x-axis.
What matrix represents a stretch of scale factor k parallel to the y-axis?
The matrix
(1 0)
(0 k)
represents a stretch of scale factor k parallel to the y-axis.
What matrix represents an enlargement of scale factor k centre the origin?
The matrix
(k 0)
(0 k)
represents an enlargement of scale factor k centre the origin.
How do you represent an anticlockwise rotation by angle theta about the origin?
The matrix
(cos(θ) -sin(θ))
(sin(θ) cos(θ))
represents an anticlockwise rotation by angle (theta) about the origin.
For 3D reflections, what represents a reflection in x = 0?
The matrix
(-1 0 0)
(0 1 0)
(0 0 1)
represents a reflection in x = 0.
For 3D reflections, what represents a reflection in y = 0?
The matrix
(1 0 0)
(0 -1 0)
(0 0 1)
represents a reflection in y = 0.
For 3D reflections, what represents a reflection in z = 0?
The matrix
(1 0 0)
(0 1 0)
(0 0 -1)
represents a reflection in z = 0.
For 3D rotations, what represents a rotation around x=0?
The matrix
(1 0 0)
(0 cos(θ) -sin(θ))
(0 sin(θ) cos(θ))
represents a rotation around the x axis.
For 3D rotations, what represents a rotation around y=0?
The matrix
(cos(θ) 0 -sin(θ))
(0 1 0)
(-sin(θ) 0 cos(θ))
represents a rotation around the y-axis.
For 3D rotations, what represents a rotation around z=0?
The matrix
(cos(θ) -sin(θ) 0)
(sin(θ) cos(θ) 0)
(0 0 1)
represents a rotation around the z-axis.
What is an invariant point?
A point which is unaffected by a transformation is known as an invariant point.
Given a transformation matrix T and position vector x, if Tx = x, then x represents an invariant point.
What is the rule for linear transformations?
For any linear transformation, (0,0) is an invariant point.
If every point on a line is mapped to itself under a transformation then what is it known as?
If every point on a line is mapped to itself under a transformation then it is known as a line of invariant points.
If every point on a line is mapped to another point one the same line then, what is it known as?
If every point on a line is mapped to another point one the same line then, it is known as an invariant line.
What is the determinant of a matrix?
If A =
(a b)
(c d)
then the determinant of a is ad - bc and is denoted |A| or det(A)
What defines a singular matrix?
If det(A) = 0, then A is called a singular matrix.
How do you find the inverse matrix?
If A =
(a b)
(c d)
then the inverse matrix is given by A⁻¹ =
1/det(A) times…
(d -b)
(-c a)
What is the rule regarding any square, non-singular matrix A?
For any square, non-singular matrix A: det(A⁻¹) = 1/(det(A))
Define an Eigenvector? Therefore, define an Eigenvalue?
A vector matrix that is scaled linearly by a non-traditional transformatory matrix is called an Eigenvector, and the scale factor is known as the Eigenvalue.
