Exam 1

Question 1

1.1 systems of linear equations

What is row equivalence in matrices?

Answer 1

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

Question 2

1.1 systems of linear equations

If the augmented matrices of two linear systems are row equivalent, what can you say about their solution set?

Answer 2

The two systems have the same solution set.

Question 3

1.2 row reduction and echelon forms

What three properties must a matrix satisfy to be in echelon form?

Reduced echelon form?

Answer 3

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 zeroes.

Reduced:

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

Question 4

1.2 row reduction and echelon forms

What is consistency?
What is uniqueness?

Answer 4

Consistent: at least one solution exists.
Uniqueness: If a solution exists, it is the only solution.

Question 5

1.2 row reduction and echelon forms

What can you say about the uniqueness of the reduced echelon form?

Answer 5

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

Question 6

1.2 row reduction and echelon forms

What is a pivot position in a matrix A?

Answer 6

A location in A that corresponds to a leading 1 in the reduced echelon form of A.

Question 7

1.2 row reduction and echelon forms

What is a pivot column in a matrix A?

Answer 7

A pivot column is a column of A that contains a pivot position.

Question 8

1.2 row reduction and echelon forms

What is a basic variable?
What is a free variable?

Answer 8

Basic: Described in terms of the free variable(s)
Free: We may choose any value for this variable.

Question 9

1.2 row reduction and echelon forms

If a linear system is consistent, what does the echelon form of a matrix look like?

Answer 9

The rightmost column of the augmented matrix is not a pivot column. e.g. there is no row of the form

[0 … 0 b]

Question 10

1.2 row reduction and echelon forms

If a linear system is consistent, what can you say about its solution set?

Answer 10

Contains either (i) a unique solution, when there are no free variables, or (ii) infinitely many solutions, when there is at least one free variable.

Question 11

1.3 Vector Equations

What is the zero vector?

Answer 11

Vector whose entries are all 0.

e.g. np.matrix(‘0; 0; 0; 0’)

Question 12

1.3 Vector Equations

What is a linear combination?

Answer 12

Given vectors v1, v2, …, vp in R^n and given scalars c1, c2, …, cp, the vector y defined by

y = c1v1 + c2v2 + … + cpvp

is a linear combination of v1, …, vp with weights c1, …, cp.

Question 13

1.3 Vector Equations

What is the relationship between the solution sets of augmented matrices and vector equations?

Answer 13

A vector equation
x1a1 + x2a2 + … + xnan = b

has the same solution set as
[a1 a2 … an b]

Question 14

1.3 Vector Equations

When can b be generated by a linear combination of a1, …, an?

Answer 14

If and only if there exists a solution to the linear system corresponding to [a1 a2 … an b]

Question 15

1.3 Vector Equations

What is Span {v1, …, vp} in terms of linear combinations?

What is this called?

Answer 15

The set of all linear combinations of v1, …, vp.

Called the subset of R^n spanned/generated by v1, …, vp.

e.g. the set of all vectors that can be written in the form
c1v1 + c2v2 + … + cpvp

Question 16

1.3 Vector Equations

How do you compute if a vector b is in Span{v1, …, vp}?

Answer 16

Ask whether the vector equation
x1v1 + x2v2 + … + xpvp = b
has a solution,

or equivalently, asking whether a system with the augmented matrix
[v1 … vp b] has a solution.

Question 17

1.3 Vector Equations

Why must the zero vector be in Span{v1, …, vp}?

Answer 17

Because no matter what each vector in a linear combination is, you can have weights such that
0v1 + … + 0vp,
which will always be the zero vector.

Question 18

1.4 The Matrix Equation Ax=b

If A is an mxn matrix, with columns a1, …, an, and if x is a vector in R^n, then what is the product of A and x?

Answer 18

The product is the linear combination of the columns of A using the corresponding entries in x as weights.

e.g. Ax = [a1 a2 … an] * np.matrix(‘x1 ; …; xn’)
= x1a1 + x2a2 + … + xnan

Question 19

1.4 The Matrix Equation Ax=b

If you have the matrix equation Ax=b, what vector equation and system of linear equations does it have the same solution set as?

Answer 19

Ax = b

x1a1 + x2a2 + … + xnan = b, where each an is a vector and each x is a component.

[a1 a2 … an b]

Question 20

1.4 The Matrix Equation Ax=b

Under what conditions does Ax=b have a solution?

Answer 20

if and only if b is a linear combination of the columns of A.

Question 21

1.4 The Matrix Equation Ax=b

Given:
Each b in R^m, the equation Ax=b has a solution

What other statements are logically equivalent?

Answer 21

1. For each b in R^m, the equation Ax=b has a solution
2. Each b in R^m is a linear combination of the columns of A
3. The columns of A span R^m
4. the coefficient matrix A has a pivot position in every row.

Question 22

1.4 The Matrix Equation Ax=b

If A is an mxn matrix, u and v are vectors in R^n, and c is a scalar, then what are some properties about how they multiply?

Answer 22

a. A(u+v) = Au + Av

b. A(cu) = c(Au)

Question 23

1.5 Solution Sets of Linear Systems

When is a system of linear equations said to be homogenous?

Answer 23

If it can be written in the form Ax = 0.

Always has at least one solution (x=0)

Question 24

1.5 Solution Sets of Linear Systems

What is the trivial solution?

Answer 24

x=0 (the zero vector in R^m).

Question 25

1.5 Solution Sets of Linear Systems

When does the homogeneous equation Ax=0 have a nontrivial solution?

Answer 25

If and only if the equation has at least one free variable.

Question 26

1.5 Solution Sets of Linear Systems

Suppose the equation Ax=b is consistent for some given b, and let p be a solution.

What is the solution set of Ax=b?

Answer 26

The set of all vectors of the form w = p + v_h, where v_h is any solution of the homogeneous equation Ax=0.