Lecture Chapter 2 Flashcards
How to A+B?
add corresponding entries
How to scalar multiple cA?
multiply each entry by c
How do you compute AB
AB = [Ab1 Ab2 … Abp]
each column of AB is a linear combination of the columns of A using weights from the corresponding entries of B
in general AB /= BA
What is the transpose of A : A^T
the columns of A^T are the rows of A
What can you say if A and B are invertible?
A^-1A=I and AA^-1=I Ax=b has unique solution x=A^-1 b A^-1 is invertible AB invertible : (AB)^-1 = B^-1 A^-1 A^T invertible : (A^T)^-1 = (A^-1)^T A row equivalent to In (any sequence of elementary row operations that reduce A to In transforms In into A^-1)
What can you say if det A = 0 ?
A is not invertible
How do you find A^-1
row reduce [A/I]
if A is row equivalent to I, [A/I] is row equivalent to [I/A^-1]
otherwise, A doesn’t have an inverse
State the Invertible Matrix Theorem (IMT)
- A invertible
- A row equivalent to In
- A has n pivot positions
- Ax=0 only has the trivial solution
- columns of A : linearly independent set
- transformation T(x) = Ax is one-to-one
- Ax=b has at least one solution
- columns of A span Rn
- T(x) = Ax maps Rn onto Rn
- nxn matrix C exists such that CA = I
- nxn matrix D exists such that AD = I
- A^T is invertible
What can you say if AB = I ?
A and B are invertible and A=B^-1 and B = A^-1
When is a linear transformation invertible?
If a transformation S exists such that S= inverse of T and S=T^-1
When is a matrix transformation invertible?
When A is invertible
What is the LU factorization?
A = LU with
L : square lower triangular matrix
U : upper triangular matrix
What is a lower triangular matrix? upper?
lower : zeros above main diagonal
upper : zeros below main diagonal
