Linear Algebra Flashcards
For two matrices A and B, is AB = 0 sufficient to say A or B = 0?
No
What is the transpose of AB?
(AB)’ = B’A’
What is a diagonal matrix?
A square matrix with aij = 0 if i ≠ j
What is the inner product of two n-vectors a and b?
a . b = a1b1 + … + anbn
What are orthogonal vectors?
a, b are orthogonal iff a . b = 0
Orthogonal = perpendicular
What is the matrix that describes anticlockwise rotation by an angle θ about the origin?
cosθ -sinθ
sinθ cosθ
What is an orthogonal matrix?
A matrix whose transpose is equal to its inverse
What is an orthogonal transformation?
A transformation that preserves the norms of transformed vectors, corresponding to an orthogonal matrix
For some scalar c and some matrix A, what is the inverse of cA?
1/c * A-1
What is the inverse of a 2x2 matrix?
A = a b
c d
A-1
= 1/|A| * d -b
-c a
How do you find the inverse of a matrix without using Gaussian elimination?
Find the determinant, find the matrix of minors, apply the signs, transpose the matrix, divide by the determinant
What are the steps for Gaussian elimination?
Set up an augmented matrix with the original matrix on the left and the identity matrix on the right
Perform EROs following this algorithm:
1) swap rows so that the top left is non zero
2) make the rest of the first column 0
3) swap so that the second diagonal entry is non zero
4) clear the rest of the second column
5) clear the top two entries of the third column
What are the definitions for the different definitesses of matrices?
A is a symmetric nxn matrix, for any n vector v
A is positive definite if v’Av > 0 and v ≠ 0
A is positive semi-definite if v’Av ≥ 0
A is negative definite if v’Av < 0 v ≠ 0
A is negative semi-definite if v’Av ≤ 0
What is an idempotent matrix?
M = M2
