Vectors and Matrices Flashcards
Scalar
Only has magnitude
Vector
Has direction and magnitude
Multiplying vectors
Multiply by a scalar constant
Adding/subtracting vectors
Add/subtract x1 and x2, then y1 and y2
Scalar product
Multiply corresponding coordinates and add up the results
v ∙ v = ?
|v|^2
u ∙ v = ?
|u||v| cos φ
Commutativity
u ∙ v = v ∙ u
Distributivity
u ∙ (v + w) = u ∙ v + u ∙ w
(au) ∙ (bv) = ?
(ab)(u ∙ v)
Linear combination
Sum of vectors by multiplying original vectors
Matrix
A table consisting of rows and columns of numbers
Square matrix
A matrix where the number of rows = number of columns
Adding/subtracting matrices
Performed element by element
Remember that you can only add/subtract matrices with the same dimensions!
Multiplication by a constant
All elements are multiplied by the constant
Multiplication by a vector
Multiply row by row using scalar product
Remember that the vector must have the same number of elements as there are columns in the matrix!
Matrix-by-matrix multiplication
Multiply element by element using scalar product (row in A by column in B)
Position vector
The vector from the origin to a point
Rotation matrix
To rotate p by an angle φ counter-clockwise about the origin, calculate p’ = Rp, where R is:
(cosφ -sinφ)
(sinφ cosφ)
Identity matrix
Iv = v
Inverse matrix
Given a square matrix A, if there is a matrix B such that BA = I, then B is the inverse of A
How can inverse matrices be used to solve simultaneous equations?
o If A-1 is known, multiply both sides by it
o Matrix multiplication is associative, so can shift brackets
o A-1A = I by definition of inverse matrix
o Iv = v by definition of identity matrix
