finals: sections 9.3 - 2.8

Question 1

what is a slack variable?

Answer 1

a nonnegative variable that is added to the smaller side of an inequality to make it an equality

Question 2

what are the steps to solve a linear programming problem using the simplex method?

Answer 2

1) restate the original problem using slack variables
2) write the augmented matrix of the equalities made in step 1
3) identify which variable to change: it is the one that has the largest negative coefficient in the augmented matrix. which one should go? look at the ratios of the largest coefficient over the coefficient of the variables that are nonzero and choose the smallest one
4) perform row operations to to get a one in the variable that we changed and zeros in the rest of that column
5) repeat the process until all coefficients in the M equation are negative, then that is the optimal solution

Question 3

what is an invertible matrix?

Answer 3

a nxn (square) matrix is said to be invertible if there is a matrix C such that
AC = I and CA = I

Question 4

how do you find the inverse of a matrix for any matrix?

Answer 4

[A I] algorithm

Question 5

how do you find the inverse of a 2x2 matrix?

Answer 5

1/(ad-bc)[[d -b],[-c a]]

Question 6

what is another name for a matrix that is not invertible?

Answer 6

singular matrix

Question 7

what is another name for a matrix that is invertible?

Answer 7

nonsingular matrix

Question 8

what is the determinant of a 2x2 matrix A?

Answer 8

det A = ad - bc

Question 9

if A is an invertible matrix, then for each b in R^n, the equation Ax = b has the ________

Answer 9

UNIQUE solution x = A^-1b

Question 10

what are the three rules for inverses?

Answer 10

(A^-1)^-1 = A
(AB)^-1 = B^-1 * A^-1
(A^T)^-1 = (A^-1)^T

Question 11

an nxn matrix A is invertible if and only if _________

Answer 11

A is row equivalent to I

Question 12

what are the statements in the Invertible Matrix Theorem?

Answer 12

a. A is an invertible matrix.
b. A is row equivalent to the n n identity matrix.
c. A has n pivot positions.
d. The equation Ax D 0 has only the trivial solution.
e. The columns of A form a linearly independent set.
f. The linear transformation x 7! Ax is one-to-one.
g. The equation Ax D b has at least one solution for each b in R^n
h. The columns of A span R^n
i. The linear transformation x 7! Ax maps R^n onto R^n
j. There is an nxn matrix C such that CA = I .
k. There is an nxn matrix D such that AD = I
l. A^T is an invertible matrix

Question 13

what is the definition of a factorization?

Answer 13

a factorization of a matrix A is an equation that expresses A as the product of two or more matrices

Question 14

what is an upper triangular matrix?

Answer 14

the matrix has stuff on the upper triangle, and zeros on the lower triangle (echelon form)
* * * *
0 * * *
0 0 * *
0 0 0 *

Question 15

what is a lower triangular matrix?

Answer 15

the matrix has stuff on the lower triangle, and zeros on the upper triangle
* 0 0 0
* * 0 0
* * * 0
* * * *

Question 16

what is the LU factorization?

Answer 16

A = LU, where A is an mxn matrix
L is an mxm lower triangular matrix with 1s on the diagonal
U is the mxn echelon form of A

Question 17

how do you find x in the equation Ax = b using LU factorization?

Answer 17

1) Ly = b
2) Ux = y

Question 18

what is a subspace in R^n?

Answer 18

any set H in R^n that has three properties:
1) the zero vector is in H
2) H is closed under vector addition
3) H is closed under scalar multiplication

Question 19

what is the column space of a matrix?

Answer 19

Col A = span{columns of matrix A}

Question 20

what is the null space?

Answer 20

Nul A is the set of all solutions of the homogeneous equation Ax = 0

Question 21

a basis for a subspace H of R^n is _________________

Answer 21

a linearly independent set in H that spans H

Question 22

what is a subspace spanned by a set?

Answer 22

given any subspace H of V, a spanning set for H is a set {v1, …, vp} such that H = span{v1, …,vp}

Question 23

what is row space?

Answer 23

Row A = span{rows of A}