1. Linear Equations in Linear Algebra

Question 1

What lies at the heart of linear algebra?

Answer 1

Systems of linear equations.

Question 2

What is a linear equation?

Answer 2

A linear equation in the variables x_1,……,x_n is an equation that can be written in the form
(a_1)(x_1) = (a_2)(x_2) + ……. (a_n)(x_n) = b
where b and the coefficients a_1, … a_n are real or complex numbers, usually known in advance.

Question 3

What is a system of linear equations?

Answer 3

A system of linear equations (or a linear system) is a collection of one or more linear equations involving the same variables - say, x_1, …..x_n

Question 4

What is the solution of a linear system?

Answer 4

A list (s_1, s_2, … s_n) of numbers that makes each equation a true statement when the values s_1, s_2, …, s_n are substituted for x_1, x_2, …, x_n respectively.

Question 5

What is a solution set?

Answer 5

The set of all possible solutions of a linear system.

Question 6

When are two linear systems said to be equivalent?

Answer 6

If they have the same solution set. That is, each solution of the first system is a solution of the second system, and each solution of the second system is a solution of the first.

Question 7

What does finding the solution set of a system of two linear equations amount to?

Answer 7

Finding the intersection of two lines.

Question 8

Name two ways in which two lines need not intersect at a single point.

Answer 8

They could be parallel (no solution)

They could be coinciding and “intersect” at every point (infinitely many solutions).

Question 9

When is a linear system said to be consistent?

Answer 9

When it has either one solution or infinitely many solutions.

Question 10

When is a linear system said to be inconsistent?

Answer 10

When it has no solution.

Question 11

What is a matrix?

Answer 11

A rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

Question 12

What is a coefficient matrix?

Answer 12

A matrix with the coefficients of each variable aligned in columns.

Question 13

What is an augmented matrix?

Answer 13

A coefficient matrix with an added column containing constants from the right sides of the equations.

Question 14

What does the size of a matrix tell us?

Answer 14

How many rows and columns it has.

Question 15

How is the size of a matrix with 3 rows and 4 columns called termed?

Answer 15

A 3 by 4 matrix (3x4 matrix)

Question 16

Describe an m x n matrix.

Answer 16

If m and n are positive integers, an m x n matrix is a rectangular array of numbers with m rows and n columns.

Question 17

What is an algorithm?

Answer 17

A systematic procedure.

Question 18

What is the basic strategy for solving linear systems?

Answer 18

To replace one system with an equivalent system (ie., one with the same solution set) that is easier to solve.

Question 19

Roughly speaking, what is the algorithm for solving a linear system?

Answer 19

Use the x_1 term in the first equation of a system to eliminate the x_1 terms in the other equations. Then use the x_2 term in the second equation to eliminate the x_2 terms in the other equations, and so one, until you finally obtain a very simple equivalent system of equations.

Question 20

How do you verify that a solution set is in fact correct?

Answer 20

By replacing the values of the corresponding variables back into the initial equations to see if they are true.

Question 21

Name and describe the three elementary row operations.

Answer 21

1. Replacement - Add to one row a multiple of another row.
2. Interchange - Interchange two rows.
3. Scaling - Multiply all entries in a row by a nonzero constant.
   It is important to note that row operations are reversible.

Question 22

When are two matrices row equivalent?

Answer 22

When there is a sequence of elementary row operations that transform one matrix into the other.

Question 23

What is a general conclusion of augmented matrices that are row equivalent?

Answer 23

They have the same solution sets.

Question 24

What are the two fundamental questions about a linear system?

Answer 24

1. Is the system consistent; does at least one solution exist?
2. If a solution exists, is it the only one?; is the solution unique?

Question 25

What is a nonzero row or columns?

Answer 25

A row or column that contains at least one nonzero entry.

Question 26

What is a leading entry?

Answer 26

A leading entry of a row refers to the leftmost nonzero entry (in a nonzero row).

Question 27

What are the three properties of matrices in echelon form and the two additional properties of matrices in reduced echelon form?

Answer 27

Echelon form

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.

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 28

What is a main difference between the uniqueness of echelon form versus reduced echelon form?

Answer 28

Any non zero matrix may be row reduced into more than one matrix in echelon form, using different sequences of row operations. However, the reduced echelon form one obtains from a matrix is unique.

Question 29

When is one matrix said to be an echelon form of another matrix?

Answer 29

When they are row equivalent.

Question 30

What is an interesting connection between reduced echelon matrices and echelon matrices?

Answer 30

When row operations on a matrix produce an echelon form, further row operations obtain the reduced echelon form do not change the position of the leading entries.

Question 31

Why don’t the additional row operations from echelon to reduced echelon change the position of the leading entries?

Answer 31

Since reduced echelon form is unique, the leading entries are always in the same positions in any echelon form obtained from a given matrix. These leading entries correspond to leading 1’s in the reduced echelon form.

Question 32

What is a pivot position?

Answer 32

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.

Question 33

What is a pivot column?

Answer 33

A column that contains a pivot position.

Question 34

What is a pivot?

Answer 34

A nonzero number in a pivot position that is used as needed to create zeros via row operation.

Question 35

Describe the 4 steps in the forward phase of the row reduction algorithm as well as the backward phase (step 5).

Answer 35

1. Begin with the leftmost nonzero column. This is a pivot column. The pivot position is at the top.
2. Select a nonzero entry in the pivot column as a pivot. If necessary, interchange rows to move this entry into the pivot position.
3. Use row replacement operations to create zeros in all positions below the pivot.
4. Cover (or ignore) the row containing the pivot position and cover all rows, if any, above it. Apply steps 1-3 to the submatrix that remains. Repeat the process until there are no more non zero rows to modify.
5. Beginning with the rightmost pivot and working upward and to the left, create zeros above each pivot. If a pivot is not 1, make it 1 by a scaling operation.

Question 36

What does the row reduction algorithm directly lead to when applied to the augmented matrix of a system?

Answer 36

an explicit description of the solution set of the linear system.

Question 37

What are basic variables?

Answer 37

The variables corresponding to pivot columns in the matrix.

Question 38

What are free variables?

Answer 38

The variables that are not pivot columns.

Question 39

When there are basic and free variables, how do we describe the solution set?

Answer 39

The solution set can be described explicitly by solving the reduced system of equations for the basic variables in terms of the free variables.

Question 40

What does a different choice of input for a free variable do to a solution?

Answer 40

Each different choice of free variable determines a (different) solution of the system, and every solution of the system is determined by a choice of the free variable.

Question 41

What is a parametric description of solution sets?

Answer 41

Solution sets in which free variables act as parameters. We make the arbitrary convention of always using free variables as the parameters for describing a solution set.

Question 42

Describe the existence and uniqueness theorem.

Answer 42

A linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column - that is, only if an echelon form of the augmented matrix has no rows in the form [0 …….. 0 b].
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 solutions, when there is at least one free variable.

Question 43

Describe the 5 steps to solving a linear system using row reduction.

Answer 43

1. Write the augmented matrix of the system.
2. Use the row reduction algorithm to obtain an equivalent matrix in echelon form. Decide whether the system is consistent. Stop here is it’s inconsistent.
3. Continue row reduction to obtain the reduced echelon form.
4. Write the system of equations corresponding to the matrix obtained in step 3.
5. Rewrite each nonzero equation from step 4 so that its one basic variable is expressed in terms of any free variable appearing in the equation.

Question 44

What is a vector?

Answer 44

An ordered list of numbers.

Question 45

What is a column vector (or simply a vector)?

Answer 45

A matrix with only one column.

Question 46

In regard to vectors, what does ℝ² denote?

Answer 46

The set of all vectors with two entries, read: r-two. The ℝ stands for the real numbers that appear as entries in the vectors, and the exponent ² indicates that each vector contains two entries.

Question 47

When are two vectors in ℝ² equal?

Answer 47

If and only if their corresponding entries are equal. They need to be corresponding entries because vectors in ℝ² are ordered pairs.

Question 48

How is the sum of two vectors in ℝ² calculated?

Answer 48

By adding the corresponding entries of both vectors.

Question 49

How is a vector multiplied by a constant?

Answer 49

The scalar multiple is the vector obtained by multiplying each entry in the vector by the constant.

Question 50

How is the geometric visualization of a vector often aided?

Answer 50

By including an arrow from the origin (0,0) to the point (x1,x2)

Question 51

What is the parallelogram rule for vector addition?

Answer 51

If u and v in ℝ² are represented as points in the plane, then u+v corresponds to the fourth vertex of the parallelogram whose other vertices are u, 0, and v.

Question 52

Describe vectors in ℝ³.

Answer 52

Vectors in ℝ³ (read r-three) are 3x1 column matrices with three entries. they are represented geometrically by points in a three-dimensional coordinate space, with arrows from the origin sometimes included for visual clarity.

Question 53

Describe vectors in ℝ^n

Answer 53

If n is a positive integer, ℝ^n (read r-n) denotes the collection of all list (or ordered n-tuples) of n real numbers, usually written as nx1 column matrices.

Question 54

What is the zero vector?

Answer 54

The vector whose entries are all zero, denoted by 0.

Question 55

Describe the 8 algebraic properties of ℝ^n for all u, v, w, in ℝ^n and all scalars c and d.

Answer 55

(i) u+v = v+u
(ii) (u+v) + w = u + (v+w)
(iii) u+0 = 0+u = u
(iv) u + (-u) = -u + u = 0
(v) c(u+v) = cu + cv
(vi) (c+d)u = cu + du
(vii) c(du) = (cd)u
(viii) 1u = u

Question 56

What is a linear combination?

Answer 56

Given vectors v_1, v_2, …. , v_p in ℝ^n and given scalars c_1, c_2,….., c_p, the theory y defined by
y = (c_1)(v_1) + … + (c_p)(v_p)
is a linear combination of v_1, …, v_p with weights c_1,…., c_p, which can be any real number including 0.

Question 57

What does a vector equation
x1a1 + x2a2 + … + xnan = b
have the same solution set as?

Answer 57

The linear system whose augmented matrix is
[a1 a2 … an b]
In particular, b can be generated by by liner combination of a1, …, an if and only if there exists a solution to the linear system corresponding to the matrix.

Question 58

What is one of the key ideas in linear algebra?

Answer 58

To study the set of all vectors that can be generated or written as a linear combination of a fixed set {v1, …, vp} of vectors.

Question 59

Describe subsets and spans.

Answer 59

If v1, …, vp are in ℝ^n, then the set of all linear combinations of v1, …, vp is denoted by Span{v1, …, vp} and is called the subset of ℝ^n spanned (or generated) by v1, …, vp. That is, Span {v1, …, vp} is the collection of al vectors that can be written in the form
c1v1 + c2v2 + … + cpvp
with c1, …., cp scalars.

Question 60

What does asking whether a vector b is in Span{v1, …, vp} amount to?

Answer 60

Asking whether the vector equation
x1v1 + x2v2 + … + xpvp = b
has a solution, or equivalently, asking whether the linear system with augmented matrix [v1 … vp b] has a solution.

Question 61

Describe the geometry of Span{v} and Span{u,v}.

Answer 61

Let v be a nonzero number in ℝ³. Then Span{v} is the set of all scalar multiples of v which is the set of points on the line in ℝ³ through v and 0.
If u and v are non zero vectors in ℝ³, with v not a multiple of u, then Span{u,v} is the plane in ℝ³ that contains u, v, and 0. In particular, Span{u, v} contains the line in ℝ³ through u and 0 and the line through v and 0.