Matrix Algebra, Linear Transformations Flashcards
What is the fundamental matrix equation?
Ax = b
What is the precondition that has to be fulfilled for an equation to have a solution?
B has to be a linear combination of the columns of A
What is the Identity Matrix I?
A matrix with 1s on the diagonal and 0s everywhere else i.e.
1 0 0
0 1 0
0 0 1
What do conditions have to be fulfilled for a transformation to be linear?
- T( u + v) = T(u) + T(v) for all u, v in the domain of T
2. T(cu) = cT(u) for all scalars c and all u in the domain of T
What is a common application of the dot product of matrix-vector multiplication?
To blur photos by replacing each pixel in face with
average of pixel intensities in its neighborhood
What are the properties of matrix multiplication?
Let A be an m x n matrix and let B and C have sizes for which the indicated sums and products are defined.
a. A(BC) = (AB)C Associative law
b. A(B+C) = AB + AC (Left distributive law)
c. (B+C)A = BA + BC (Right distributive law)
d. r(AB) = (rA)B = (rB)A for any scalar r
e. ImA = A = AIn (Identity for matrix multiplication)
In general is AB = BA ?
No
What is the relationship between matrix inverse and identity matrix?
A-1A = I and AA-1 = I
a matrix multiplied with its inverse is the identity matrix
What’s the name of a non invertible matrix?
singular matrix
When is a matrix non - ivertible?
when its determinant is not equal to 0
How do you calculate the determinant of a 2x2 matrix?
a b
c d
det A = ad - bc
What is the transpose of a matrix?
given a m x n matrix the transpose of A is n x m matrix i.e. a b c d transpose a c b d
