LINEAR ALGEBRA FINAL TEST
QUESTIONS AND ANSWERS
rank - Answer-dimension of the column space/number of pivot columns
range(A) - Answer-col(A)
rank nullity - Answer-n = rank + dim(ker)
injective/one to one - Answer-rank = # cols (n), pivot in every col
surjective/onto - Answer-rank = # rows (m), pivot in every row
basis - Answer-spanning set and linearly independent
orthogonality means - Answer-linearly independent
inverse of a 2x2 matrix - Answer-1/det [d -b -c a]
A is invertible - Answer-if A is a square, 0 is not an eigenvalue, det is not 0
A is diagonalizable - Answer-if A is square, has n lin ind eigenvectors, geo mult = alg
mult for every eigenvector, D = P^-1AP exists
Area of a parallelogram spans A - Answer-abs value of det(A)
Volume of a parallelepiped spans A - Answer-abs value det(A)
area/volume of transformation A - Answer-abs value det(A) * area/volume of shape
orthogonal matrix - Answer-square invertible matrix A, the columns and rows are
orthonormal, where A^T = A^-1 and A^TA = Identity matrix
(AB)^-1 - Answer-B^-1A^-1
det(B^-1) - Answer-1/det(B)
similar matrices - Answer-have the same eigenvalues and multiplicities
least squares solution - Answer-A^TAx = A^Tb
counterclockwise rotation matrix - Answer-cos -sin
sin cos
QUESTIONS AND ANSWERS
rank - Answer-dimension of the column space/number of pivot columns
range(A) - Answer-col(A)
rank nullity - Answer-n = rank + dim(ker)
injective/one to one - Answer-rank = # cols (n), pivot in every col
surjective/onto - Answer-rank = # rows (m), pivot in every row
basis - Answer-spanning set and linearly independent
orthogonality means - Answer-linearly independent
inverse of a 2x2 matrix - Answer-1/det [d -b -c a]
A is invertible - Answer-if A is a square, 0 is not an eigenvalue, det is not 0
A is diagonalizable - Answer-if A is square, has n lin ind eigenvectors, geo mult = alg
mult for every eigenvector, D = P^-1AP exists
Area of a parallelogram spans A - Answer-abs value of det(A)
Volume of a parallelepiped spans A - Answer-abs value det(A)
area/volume of transformation A - Answer-abs value det(A) * area/volume of shape
orthogonal matrix - Answer-square invertible matrix A, the columns and rows are
orthonormal, where A^T = A^-1 and A^TA = Identity matrix
(AB)^-1 - Answer-B^-1A^-1
det(B^-1) - Answer-1/det(B)
similar matrices - Answer-have the same eigenvalues and multiplicities
least squares solution - Answer-A^TAx = A^Tb
counterclockwise rotation matrix - Answer-cos -sin
sin cos
