1.2 Gaussian Elimination

Question 1

numerical analysis

Answer 1

development and analysis of methods to approximate the solutions of mathematical problems

Question 2

reduced row echelon form

Answer 2

a matrix is in reduced row echelon form if and only if it is in row echelon form, all its pivots are equal to 1 and the pivots are the only non-zero entries of the basic columns

Question 3

leading 1

Answer 3

the first non-zero entry in a row of a matrix that is in reduced row-echelon form

Question 4

row echelon form

Answer 4

the matrix in echelon form can contain any values as its non-zero entries,

Question 5

leading variables

Answer 5

a variable that corresponds to a column in a reduced row-echelon form (RREF) matrix that contains a leading one

Question 6

free variables

Answer 6

a variable that has no limitations or restrictions

Question 7

elimination procedure

Answer 7

a systematic way to solve linear equations by removing one variable at a time

Question 8

Gauss–Jordan elimination

Answer 8

creating an augmented matrix of both sides of our equations, changing this matrix into reduced row echelon form, and then finishing up the problem to find our solution.

Question 9

forward phase

Answer 9

Reducing a matrix to echelon form

Question 10

backward phase

Answer 10

reducing a matrix to a reduced echelon form

Question 11

homogeneous

Answer 11

a system where all the constants on the right side of the equals sign are zero

Question 12

trivial solution

Answer 12

a solution to a system of linear equations where all variables are equal to zero

Question 13

nontrivial solutions

Answer 13

a non-zero solution to a matrix equation

Question 14

back-substitution

Answer 14

values are substituted back into the equations starting from the last variable and moving towards the first variable to find the solution

Question 15

pivot positions

Answer 15

a location that corresponds to a leading 1 in the matrix’s reduced echelon form

Question 16

pivot column

Answer 16

a column in a matrix that contains a pivot, which is the first nonzero entry in a row of a matrix in row-echelon form

Question 17

roundoff

Answer 17

the difference between an approximation of a number used in computation and its exact (correct) value

Question 18

unstable

Answer 18

when successive calculations may degrade an answer to a degree that makes it useless