01 Linear Equations in Linear Algebra

Question 1

consistent system of linear equations

Answer 1

A system of linear equations is said to be consistent if it has either one solution or infinitely many solutions; a system is inconsistent if it has no solution.

Question 2

Elementary row operations

Answer 2

1. Replacement
2. Interchange
3. Scaling

Question 3

echelon form

Answer 3

A rectangular matrix is in echelon form (or row echelon form) if it has the following three properties:

1. All nonzero rows are above any rows of all zeros
2. Each leading entry of a row is in a column to the right of the leading entry of the row above it
3. All entries in a column below a leading entry are zeros

If a matrix in echelon form satisfies the following additional conditions, then it is in reduced echelon form:

4. The leading entry in each nonzero row is 1
5. Each leading 1 is the only nonzero entry in its column

Question 4

Each matrix is row equivalent to one and only one _____

Answer 4

Each matrix is row equivalent to one and only one reduced echelon matrix

Question 5

RREF

Answer 5

Reduced Row Echelon Form

Question 6

pivot position, pivot column

Answer 6

A pivot position in a matrix A is a location in A that corresponds to a leading 1 in the reduced echelon form of A. A pivot column is column of A that contains a pivot position.

Question 7

RREF places each basic variable in ___ equation

Answer 7

RREF places each basic variable in one and only one equation

Question 8

The variables corresponding to pivot columns in the matrix are called ____. The other variables are called ____.

Answer 8

The variables corresponding to pivot columns in the matrix are called basic variables. The other variables are called free variables.

Question 9

Row Reduction Algorithm

Answer 9

…

Question 10

The solution is not unique because there are ___ variables.

Answer 10

The solution is not unique because there are free variables.

Question 11

Existence and uniqueness theorem

Answer 11

A linear system is consistent if and only if the rightmost column of the augmented matrix not a pivot column.

Question 12

Span

Answer 12

If v1, …, vp are in Rn, then the set of all linear combinations of v1, …, vp is denoted by Span{v1, …, vp} and is called the subset of Rn spanned by v1, …, vp.

Question 13

Asking whether a vector b is in Span {v1, …, vp} amounts to asking whether_____

Answer 13

Asking whether a vector b is in Span {v1, …, vp} amounts to asking whether the linear system with augmented matrix [v1 … vp b] has a solution.

Question 14

The equation Ax = b has a solution if and only if ___

Answer 14

The equation Ax = b has a solution if and only if b is a linear combination of the columns of A.

Question 15

Let A be and m x n matrix. Then the following statements are logically equivalent. That is, for a particular A, either they are all true statements or they are all false:

Answer 15

a. For each b in Rm, the equation Ax = b has a solution
b. Each b in Rm is a linear combination of the columns of A
c. The columns of A span Rm.
d. A has a pivot position in every row.

Question 16

A system of linear equations is said to be homogeneous if it can be written in the form ____

Answer 16

Ax = 0.

Question 17

For a homogeneous system of linear equations, the important question is whether there exists a ___

Answer 17

For a homogeneous system of linear equations, the important question is whether there exists a nontrivial solution, that is, a nonzero vector x that satisfies Ax = 0.

Question 18

The homogeneous equation Ax = 0 has a nontrivial solution if and only if ____

Answer 18

The homogeneous equation Ax = 0 has a nontrivial solution if and only if the equation has at least one free variable.

Question 19

Suppose the equation Ax = b is consistent for some 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 + v_h, where v_h is any solution of ___

Answer 19

Suppose the equation Ax = b is consistent for some 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 + v_h, where v_h is any solution of the homogeneous equation Ax = 0.

Question 20

An indexed set of vectors {v1, …, vp} in Rn is said to be linearly independent if ____

Answer 20

An indexed set of vectors {v1, …, vp} in Rn is said to be linearly independent if the vector equation

x1vi + x2vi + … + xpvp = 0

has only the trivial solution.

Question 21

The ___ of a matrix A are linearly independent if and only if the equation ___ has only the trivial solution

Answer 21

The columns of a matrix A are linearly independent if and only if the equation Ax = 0 has only the trivial solution

Question 22

Characterization of Linearly Dependent Sets

Answer 22

An indexed set S = {v1, …, vp} of two or more vectors is linearly dependent if and only if at least one of the vectors in S is linear combination of the others. In fact, if S is linearly dependent and v1 != 0, then some vj (with j > 1) is a linear combination of the preceding vectors, v1, …, v_{j-1}.

Question 23

Every vector in a linearly dependent set ___ a linear combination of the other vectors.

Answer 23

is not necessarily

Question 24

If a set contains more vectors than there are entries in each vector, then the set is _____.

Answer 24

If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. That is, any set {v1, …, vp} in Rn is linearly dependent if p > n.

Question 25

If a set S = {v1, …, vp} in Rn contains the zero vector, then the set is _____.

Answer 25

If a set S = {v1, …, vp} in Rn contains the zero vector, then the set is linearly dependent.

Question 26

A transformation T from Rn to Rm is a rule that assigns to each vector x in RN A VECTOR T(x) in Rm. The set Rn is called the ____ of T, and Rm is called the ____ of T. For x in Rn, the vector T(x) in Rm is called the ____ of x (under the action of T). The set off all images T(x) is called the ____ of T.

Answer 26

A transformation T from Rn to Rm is a rule that assigns to each vector x in RN A VECTOR T(x) in Rm. The set Rn is called the domain of T, and Rm is called the codomain of T. For x in Rn, the vector T(x) in Rm is called the image of x (under the action of T). The set off all images T(x) is called the range of T.

Question 27

A transformation T is linear if:

Answer 27

A transformation T is linear if:

1. T(u + v) = T(u) + T(v) for all u, v in the domain of T
2. T(cu) = cT(u) for all scalars c and all u in the domain of T

Question 28

Let T : Rn -> Rm be a linear transformation. Then there exists a unique matrix A such that

T(x) = _____

In fact, A is the m x n matrix whose jth column is the vector T(e_j), where e_j is the jth column of the identity matrix in Rn:

A = _____

Answer 28

Let T : Rn -> Rm be a linear transformation. Then there exists a unique matrix A such that

T(x) = Ax for all x in Rn

In fact, A is the m x n matrix whose jth column is the vector T(e_j), where e_j is the jth column of the identity matrix in Rn:

A = [T(e₁) … T(e_n)]

The matrix A is called the standard matrix for the linear transformation T.

Question 29

A mapping T : Rn -> Rm is said to be onto Rm if ____

Answer 29

A mapping T : Rn -> Rm is said to be onto Rm if each b in Rm is the image of at least on x in Rn (= when the range of T is all of the codomain Rm).

Question 30

A mapping T : Rn -> Rm is said to be one-to-one if _____

Answer 30

A mapping T : Rn -> Rm is said to be one-to-one if each b in Rm is the image of at most one x in Rn.

Question 31

T is one-to-one if and only if ____

Answer 31

T is one-to-one if and only if T(x) = 0 has only the trivial solution

Question 32

T maps Rn onto Rm if and only if ____

Answer 32

T maps Rn onto Rm if and only if the columns of A span Rm