MATH 21 - LINEAR ALGEBRA: FINAL
(SECTIONS 4.1-6.2) EXAM QUESTIONS
WITH CORRECT ANSWERS
when is A invertible? - Answer-when det(A) does not equal 0
what is det(A^-1) equal to? - Answer-1/det(A)
what is the cofactor expansion equation? - Answer-Cij(A) = (-1)^i+j det(A)
what is det(A^Tr) also equal to? - Answer-det(A)
what are the elementary row operation rules when solving determinants? - Answer--
switching: det(A) -> -det(A)
- scaling: det(cA) -> cdet(A)
- elimination: det(A) -> det(A)
what happens if two rows are identical? - Answer-then det(A)=0
when is a subset W a subspace of V? - Answer-- when you can express the vector
space as a span of vectors
- can we write every element of W as a linear combo of some elements?
what do you do when you get W = span (v1 ... vk) +Po - Answer-see if Po is in the span
of v1 ... vk by setting up an augmented matrix
how do you compute the dimension of a finitely generated vector space? - Answer--
create a matrix and use gaussian elimination to get echelon form
- number of columns with a pivot position is the dim
how do you compute the dim(null(A))? - Answer-- create a matrix and use gaussian elim
to get reduced echelon form
- solve the linear system
- dim(null(A)) = number of vectors in null(A)
given a sequence B, determine if it is a basis - Answer-- B must be in V (either set up
matrix [V | B] or see if every vector in B satisfy the restriction in V
- B must be linearly independent
- [B | V] must be consistent
- dimV = length of B
what is col(A) and how do you find it? - Answer-- column space of A
- span (pivot columns of A)
(SECTIONS 4.1-6.2) EXAM QUESTIONS
WITH CORRECT ANSWERS
when is A invertible? - Answer-when det(A) does not equal 0
what is det(A^-1) equal to? - Answer-1/det(A)
what is the cofactor expansion equation? - Answer-Cij(A) = (-1)^i+j det(A)
what is det(A^Tr) also equal to? - Answer-det(A)
what are the elementary row operation rules when solving determinants? - Answer--
switching: det(A) -> -det(A)
- scaling: det(cA) -> cdet(A)
- elimination: det(A) -> det(A)
what happens if two rows are identical? - Answer-then det(A)=0
when is a subset W a subspace of V? - Answer-- when you can express the vector
space as a span of vectors
- can we write every element of W as a linear combo of some elements?
what do you do when you get W = span (v1 ... vk) +Po - Answer-see if Po is in the span
of v1 ... vk by setting up an augmented matrix
how do you compute the dimension of a finitely generated vector space? - Answer--
create a matrix and use gaussian elimination to get echelon form
- number of columns with a pivot position is the dim
how do you compute the dim(null(A))? - Answer-- create a matrix and use gaussian elim
to get reduced echelon form
- solve the linear system
- dim(null(A)) = number of vectors in null(A)
given a sequence B, determine if it is a basis - Answer-- B must be in V (either set up
matrix [V | B] or see if every vector in B satisfy the restriction in V
- B must be linearly independent
- [B | V] must be consistent
- dimV = length of B
what is col(A) and how do you find it? - Answer-- column space of A
- span (pivot columns of A)
