Chapter 1

Question 1

augmented matrix

Answer 1

the matrix that contains all of the numerical information contained in the system, including constants unassociated with variables

Question 2

coefficient matrix

Answer 2

the matrix which contains the coefficients of the variables in a linear system

Question 3

square matrix

Answer 3

if the number of rows of a matrix A equals the number of columns

Question 4

diagonal matrix

Answer 4

a square matrix organized such that all entries above and below the main diagonal are zero

Question 5

upper triangular matrix

Answer 5

square matrix organized sucht that all its entries below the main diagonal are zero

Question 6

lower triangular matrix

Answer 6

square matrix organized sucht that all its entries above the main diagonal are zero

Question 7

zero matrix

Answer 7

matrix with all entries = 0

Question 8

vector

Answer 8

matrix with only one row/column; usually indicates a column vector

Question 9

row/column vector

Answer 9

vector that is a single row/column

Question 10

vector space, and how to denote

Answer 10

the set of all column vectors with n components, denoted with big R^n

Question 11

standard representation of vectors

Answer 11

representing a vector within the Cartesian coordinate plane as an arrow (directed line segment) from the origin to the point (x, y). Default representation.

Question 12

alternatives to standard representation

Answer 12

Can translate/shift the vector in the plane and connect some point (a, b) to the point (a + x, b + y).

Question 13

gauss-jordan method

Answer 13

We proceed from equation to equation, from top to bottom.

Suppose we get to the ith equation. Let Xj be the leading variable of the
system consisting of the ith and all the subsequent equations.

• If Xj does not appear in the ith equation, swap the ith equation with the first equation below that does contain Xj.

• Suppose the coefficient of Xj in the ith equation is c; thus this equation is of
the form cxj + … = … Divide the ith equation by c.

• Eliminate xj from all the other equations, above and below the ith, by subtracting suitable multiples of the ith equation from the others.

Now proceed to the next equation. If an equation zero = nonzero emerges in this process, then the system fails to have solutions; the system is inconsistent.

When you are through without encountering an inconsistency, solve each
equation for its leading variable. You may choose the nonleading variables freely; the leading variables are then determined by these choices.

Question 14

inconsistent system

Answer 14

a system of equations with no solutions

Question 15

consistent system

Answer 15

a system of equations that has at least one solution

Question 16

reduced row-echelon form

Answer 16

A matrix is in reduced row-echelon form if it satisfies all of the following conditions:

a. If a row has nonzero entries, then the first nonzero entry is a 1, called the leading 1 (or pivot) in this row.
b. If a column contains a leading 1, then all the other entries in that column are 0.
c. If a row contains a leading 1, then each row above it contains a leading 1 further to the left.

Condition c implies that rows of 0’s, if any, appear at the bottom of the matrix.

Question 17

types of elementary row operations

Answer 17

• divide a row by a nonzero scalar
• subtract a multiple of a row from another row
• swap two rows

Question 18

rank

Answer 18

the number of leading 1s in rref(A), denoted rank(A)

Question 19

matrix

Answer 19

a rectangular array of numbers

Question 20

possible number of solutions in a linear system

Answer 20

infinitely many solutions
exactly one solution
no solutions

Question 21

number of equations (n) vs number of unknowns (m) and their impact on solutions

Answer 21

If a linear system as exactly one solution, then there must be at least as many equations as there are variables/unknowns. (m < m has either no solutions or infinitely many solutions.

Question 22

when systems of n equations in n variables have a unique solution

Answer 22

only iff the rank of the coefficient matrix A is n

Question 23

dot product of vectors v and w, and when possible

Answer 23

v1w1 + … vnwn. Must have same number of entries.

Question 24

product of matrix M and vector V, and when possible

Answer 24

If M is n x m, then vector must be of length m (have as many rows as M has columns). The result will be a vector of height/rowcount n and width/columncount 1.

Two options:

• entry i of result will be the dot product of M’s row i with V
• result will be the sum of each product of V’s single-value entries with M’s column vectors.

It’s the linear combination of the columns of M with the components of V as the coefficients, if you get that reference. Which you should.

Question 25

linear combination

Answer 25

a vector V in R^n is a linear combination of the vectors v1..vm in R^n if there exists scalars x1…xm such that

V = x1v1 + … xm * vm. In other words, a vector is a linear combination of two other vectors if there are scalars x and y such that the vector equals = x * first vector + y * second vector. Add z for a third candidate combined vector, and so forth.

Question 26

algebraic rules for matrix * vector

Answer 26

M(vector x + vector y) = M(vector x) + M(vector y)

M(k * vector x) = k(M * vector x)

Question 27

matrix form of a linear system

Answer 27

can write augmented matrix of a linear system [ M | V] as Mv = V. That’s the matrix form of a linear system.