Chapter 1

Question 1

System of Linear equations

Answer 1

One or more linear equations involving the same variables

Question 2

Solution to a system of linear equations in x1, x2, … , xn

Answer 2

A list of real (or complex) numbers (s1, s2, …, sn) that satisfies every equation in the system

Question 3

Solution set

Answer 3

The set of all possible solutions to a system of linear equations

Question 4

Equivalent systems

Answer 4

Two systems that have the same solution set

Question 5

Consistent system

Answer 5

One that has a solution (a system is inconsistent if no solution exists)

Question 6

What operations do not change the solution set of a linear system?

Answer 6

1: interchanging two equations
2: multiplying an equation by a nonzero constant
3: replacing one equation with the sum of itself and a multiple of another equation in the system.

Question 7

How many solutions can a system of linear equations have?

Answer 7

1: no solution (inconsistent)
2: exactly one solution
3: infinitely many solutions

Question 8

What does the notation: m x n matrix , denote?

Answer 8

A rectangular array of numbers with m rows and n columns

Question 9

What are the elementary row operations?

Answer 9

1: interchange- interchange two rows
2: scaling - multiply a row by a nonzero constant
3: replacement- replace one row with the sum of itself and a multiple of another row

Question 10

When are two matrices Row equivalent?

Answer 10

Two matrices are Row equivalent if there is a series of elementary row operations that transforms one into the other.

Question 11

Leading entry

Answer 11

The leftmost nonzero entry in the row of a matrix

Question 12

A matrix is in row echelon form if:

Answer 12

1 rows of all zeros are at the bottom
2 each leading entry is in a column to the right of the leading entry in the row above it
3 all columns entries below a leading entry are zero

Question 13

A matrix is in reduced row echelon form if:

Answer 13

The matrix is in row echelon form AND

1: the leading entry in each row is a 1
2: each leading 1 is the only nonzero entry in its column

Question 14

How many reduced row echelon forms is a matrix Row equivalent to?

Answer 14

One and only one

Question 15

Pivot position

Answer 15

A position of a leading entry in a matrix in a row echelon form

Question 16

Pivot column

Answer 16

Column containing a pivot position

Question 17

Pivot

Answer 17

A nonzero entry in a pivot position

Question 18

How is the Gaussian elimination algorithm performed?

Answer 18

1: start with the leftmost nonzero column- a pivot column
2: select a nonzero entry in the pivot column to be the pivot. If necessary, interchange rows to move the pivot to the top of the pivot column (into the pivot position)
3: use replacement to create zeros in all position below the pivot
4: cover the row containing the pivot and all rows above it. Apply steps 1-3 to the remaining submatrix. Repeat until the matrix is in RE form
5: beginning with the rightmost pivot and moving up and left, use row replacement to create zeros above each pivot and scale each pivot to 1

Question 19

Associated systems

Answer 19

System of linear equations represented by an augmented matrix

Question 20

Basic variable

Answer 20

Correspond to pivot columns

Question 21

Free variables

Answer 21

Correspond to columns without a pivot

Question 22

General solution to a linear system (parametric form)

Answer 22

Expresses basic variables in terms of the free variables (which act as parameters)

Question 23

What is the existence and uniqueness theorem

Answer 23

A linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column- that is, if and only if an Row echelon form has no row of the form [00…0b], with b nonzero. If the system is consistent, then the solution set contains either

i: a unique solution, when there are no free variables, or
ii: infinitely many solution, where there is a least one free variable

Question 24

What are valid operations to perform on a vector?

Answer 24

Vector addition-(u+v computed component-wise)
Scalar multiplication (ku component-wise scaling for k is a real number)

Question 25

Linear combination

Answer 25

The vector u is a linear combination of v1, …, vn if there are scalars c1,…,cp such that u= c1v1+…+cpvp

Question 26

What is span?

Answer 26

The subset of R^n spanned by v1,…,va, denoted Span{v1,…,vp}, is the set of all linear combinations of v1,…,vp. That is,
Span{v1,…, vp}= { c1v1 + … + cpvp | c1, … , cp are real numbers)

Question 27

Let A be an m x n matrix with columns a1, a2, …, an and let x be an element of R^n. THen the matrix-vector product Ax is:

Answer 27

Ax = x1a1 + x2a2 + … + xnan,

the linear combination of the n columns of A using the n entries of x as weights

Question 28

Theorem 1.4

Answer 28

Let A be an m x n matrix. The following are equivalent:

i) for each b in R^m, the equation Ax = b has a solution.
ii) each b in R^m in a linear combination of the coumns of A
iii) the columns of A span R^m
iv) A has a pivot in every row

Question 29

Homogeneous system

Answer 29

System of linear equations that can be written as:

Ax = the zero vector

Question 30

How are the solution sets for the system Ax = b and the associated homogeneous system Ax = zero vector related?

Answer 30

All solutions to the system Ax = b have the form x = p + su1 + tu2, where
1: p is a specific solution to the system
Ax = b
2: any vector of the form su1 + tu2 is a solution for the homogeneous system Ax=0

Question 31

What are the properties of the Matrix-Vector product? (Theorem 5)

Answer 31

If A is an m x n matrix, with vectors u and v elements of R^n, and a scalar c is a real number, then

a) A(u+v) = Au + Av ; and
b) A(cu) = cA(u)

Question 32

Theorem 6

Answer 32

Suppose the equation Ax =b is consistent for a given b, and let p be a solution. Then the solution set of Ax = b is the set of all vectors of the form w = p + vh where vh is any solution to the homogeneous equation Ax =0

Question 33

Linear independence

Answer 33

A set of vectors {v1, … , vp} in R^n is called linearly independent if the equation
x1v1 + x2v2 + … + xpvp = 0
has only the trivial solution. THe set is linearly dependent otherwise, i.e. if there are weights c1, … , cp, not all zero, such that
c1v1 + c2v2 + cpvp = 0

Question 34

What is the characterization of linear dependence? (Theorem 7)

Answer 34

A set of vectors {v1, … , vp} with p>1, is linearly dependent if and only if at least one of the vectors in the set is a linear combination of the others

Question 35

What is a transformation?

Answer 35

A transformation (or function or mapping) T from R^n to R^m is a rule that assigns to each vector in R^n one and only one vector T(x) in R^m

Question 36

When is a transformation said to be onto?

Answer 36

A transformation T: R^n -> R^m is said to be onto R^m (surjective) if each b in R^m is the image of at least one x in R^n.

Question 37

When is a transformation said to be one-to-one?

Answer 37

A transformation T: R^n -> R^m is said to be one-to-one (injective) if each b in R^m is the image of at most one x in R^n.

Question 38

When is a transformation said to be linear?

Answer 38

A transformation T is linear if, for all vectors u and v in the domain and all scalars k,
i) T(u+v) = T(u) + T(v) , and
ii) T(ku) =kT(u)
(We say that T preserves vector addition and scalar multiplication)

Question 39

What are the properties of a linear transformation?

Answer 39

If T is a linear transformation, then
i) T(0) = 0
ii) T(cu + dv) = cT(u) + dT(v) for all vectors u and v in the domain and scalars c and d
(Linear transformations preserve the zero vector and all linear combinations)

Question 40

Theorem 1.10

Answer 40

Let T: R^n -> R^m be a linear transformation. Then there exists a unique matrix A such that T(x) = Ax for all x in R^n. In fact, A is the m x n matrix [T(e1) T(e2) … T(en)], where ej is the jth column of the n x n identity matrix
(Every linear transformation is a matrix transformation. Also, the columns of the matrix are the images of the unit vectors)

Question 41

Theorem 1.12

Answer 41

Let T : R^n -> R^m be a linear transformation and let A be the standard matrix for T. Then,

a) T maps R^n onto R^m if and only if the columns of A span R6m
b) T is one-to-one if and only if the coumns of A are linearly independent