Chapter 1 – Key Concepts

Question 1

What are the possible solutions for a system of linear equations?

Answer 1

1. No solution, or
2. exactly 1 solution, or
3. infinitely many solutions.

Question 2

What are the elementary row operations?

Answer 2

1. (Replacement) Replace one row by the sum of itself and a multiple of another row.
2. (Interchange) Interchange two rows.
3. (Scaling) Multiply all entries in a row by a nonzero constant.

Question 3

What is the relationship between row-equivalent augmented matrices and the solution sets for the two corresponding linear systems?

Answer 3

If the augmented matrices of two linear systems are row equivalent, then the two systems have the same solution set.

Question 4

What are the two fundamental questions about linear systems that are answered by linear algebra?

Answer 4

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

Question 5

What are the definitions for row echelon form (REF) and reduced row echelon form (RREF)?

Answer 5

A rectangular matrix is in REF if:

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.

It is in RREF if it’s in REF and:

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

Question 6

What is the relationship between row equivalence and reduced row echelon form?

Answer 6

If A and B are row equivalent, they will have the exact same RREF.

Question 7

What’s a pivot position? A pivot column?

Answer 7

A pivot position is a location in a matrix A that corresponds to a leading 1 in RREF(A). A pivot column is a column of A which has a pivot position.

Question 8

What is the algorithm to put a matrix in RREF?

Answer 8

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 nonzero 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 9

What is the existence and uniqueness theorem for linear systems?

Answer 9

A linear system is consistent iff the rightmost column in its augmented matrix is not a pivot: nothing of the form [0 … 0 b]. If a system is consistent, the solution set is either unique if there are no free variables or it is an infinite set of solutions.

Question 10

How is row reduction used to solve a linear system?

Answer 10

1. Write the augmented matrix of the system.
2. Use the row reduction algorithm to obtain an equivalent augmented matrix in echelon form. Decide whether the system is consistent. If there is no solution, stop; otherwise, go to the next step.
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 variables appearing in the equation.

Question 11

What is the parallelogram rule for addition?

Answer 11

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 12

What are the algebraic properties of ℝⁿ?

Answer 12

For all u, v, w in ℝⁿ and all scalars c and d:

(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

Question 13

What’s the relationship between a vector equation and an augmented matrix?

Answer 13

A vector equation x₁a₁ + … + x_na_n = b has the same solution set {x₁ … x_n} as the augmented matrix [a₁ … a_n b]. In particular, b is a linear combination of a₁ … a_n iff there exists a solution to the linear system corresponding to the augmented matrix [a₁ … a_n b].

Question 14

What is the span of a given set of vectors?

Answer 14

For v₁, …, v_p ∈ ℝⁿ, then the set of all linear combinations of v₁, …, v_p is denoted Span{v₁, …, v_p} and is called the subset of ℝⁿ spanned (or generated) by v₁, …, v_p. Span{v₁, …, v_p} is the collection of all vectors that can be written as c₁v₁ + … + c_pv_p for c₁, …, c_p ∈ ℝ.

Question 15

What is the definition of the product of a matrix A and a vector x?

Answer 15

If A is an m×n matrix, with columns a₁, …, a_n, and if x is in ℝⁿ, then the product of A and x, denoted by Ax, is the linear combination of the columns of A using the corresponding entries in x as weights; that is, Ax = x₁a₁ + … x_na_n.

Question 16

What is the relationship between the solution sets of Ax = b, x₁a₁ + … + x_na_n = b, and the system of linear equations whose augmented matrix is [a₁ … a_n b]?

Answer 16

The solution sets for all three of these expressions are equivalent.

Question 17

What must be true about b in terms of its relation to the columns of A, if Ax = b has a solution?

Answer 17

b must be a linear combination of the columns of A.

Question 18

∀ b ∈ ℝ^m, Ax = b has a solution. If this is true, what can we say about the following:

1. The relation between any b ∈ ℝ^m and the columns of A as vectors.
2. The span of the columns of A.
3. The pivot positions of A.

Answer 18

The following must be all true or all false:

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

Question 19

What is the row-vector rule for computing Ax?

Answer 19

If the product Ax is defined, then the ith entry in Ax is the sum of the products of corresponding entries from row i of A and from the vector x.

Question 20

What are the basic algebraic properties for A is an m×n matrix, u and v are vectors in ℝⁿ, and c is a scalar?

a. A(u + v) = ?
b. A(cu) = ?

Answer 20

a. A(u + v) = Au + Av;
b. A(cu) = c(Au).

Question 21

What must be true about the homogeneous equation Ax = 0 if it has a nontrivial solution?

Answer 21

There must be a free variable in the equation.

Question 22

Suppose the equation Ax = b is consistent for some given b, and let p be a solution. Describe the solution set in terms of the solution set to the homogeneous equation Ax = 0.

Answer 22

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 23

What are the steps for writing a solution set (of a consistent system) in parametric vector form?

Answer 23

1. Row reduce the augmented matrix to RREF. 2. Express each basic variable in terms of any free variables appearing in an equation. 3. Write a typical solution 𝐱 as a vector whose entries depend on the free variables, if any. 4. Decompose 𝐱 into a linear combination of vectors (with numeric entries) using the free variables as parameters.

Question 24

If an indexed set of vectors {v₁, …, v_p} in ℝⁿ is linearly independent, what must be true about the solution set of the following vector equation?

x₁v₁ + … + x_pv_p = 0

And if {v₁, …, v_p} is linearly dependent?

Answer 24

If linearly independent, the equation must have only the trivial solution x = 0. If linearly dependent, there must exist some c₁, …, c_p ∈ ℝ not all zero such that ∑c_iv_i = 0.

Question 25

What must be true about the columns of A if the equation Ax = 0 has only the trivial solution?

Answer 25

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

Question 26

A set of two vectors {v₁, v₂} is linearly dependent if what relationship exists between the two? And if the set is linearly independent?

Answer 26

If it is dependent, v₁ = cv₂ for some c ∈ ℝ where c ≠ 0. If the set is linearly independent, then v₁ is not a multiple of v₂.

Question 27

If S = {v₁, …, v_p} has at least one vector that is a multiple of other vectors in S, is the set linearly independent or dependent? In other words, say v₁ ≠ 0, then some v_j for j > 1 is a linear combination of the preceding vectors v₁, …, v_j-1.

Answer 27

Linearly dependent.

Question 28

If a set contains more vectors than there are entries in each vector, is it linearly independent or dependent?

Answer 28

The set is linearly dependent. That is, any set {v₁, …, v_p} in ℝⁿ is linearly dependent if p > n.

Question 29

If a set contains the zero vector, is it linearly independent or dependent?

Answer 29

Dependent.

Question 30

What two conditions must hold if a transformation (or mapping) T is linear?

Answer 30

(i) T(u + v) = T(u) + T(v) for all u, v in the domain of T.
(ii) T(cu) = cT(u) for all c ∈ ℝ and all u in the domain of T.

Question 31

If T is a linear transformation, evaluate the following:

1. T(0)
2. T(cv + du) for c, d ∈ ℝ.

Answer 31

1. T(0) = 0
2. T(cv + du) = cT(v) + dT(u)

Question 32

If T is a linear transformation, evaluate the superposition principle:

T(c₁v₁ + … + c_pv_p)

Answer 32

T(c₁v₁ + … + c_pv_p) = c₁T(v₁) + … + c_pT(v_p)

Question 33

What is the relationship between linear transformations and matrices?

Answer 33

If T: ℝ^m→ℝⁿ, there exists a unique n×m matrix A (the std. matrix of T) s.t. T(x) = Ax and A = [T(e₁) … T(e_m)].

Question 34

A transformation is onto if what?

Answer 34

A mapping T: ℝⁿ→ℝ^m is said to be onto ℝ^m if each b in ℝ^m is the image of at least one x in ℝⁿ.

Question 35

A transformation is one-to-one if what?

Answer 35

A mapping T: ℝⁿ→ℝ^m is said to be one-to-one if each b in ℝ^m is the image of at most one x in ℝⁿ.

Question 36

If T: ℝⁿ→ℝ^m is a one-to-one linear transformation, what must be true about the equation T(x) = 0?

Answer 36

It must have only the trivial solution x = 0.

Question 37

Let T: ℝⁿ→ℝ^m be a linear transformation with a standard matrix A. If T is onto, what must be true about the columns of A? What about if T is one-to-one – what must be true about the columns then?

Answer 37

a. T maps ℝⁿ onto ℝ^m iff the columns of A span ℝⁿ.
b. T is one-to-one iff the columns of A are linearly independent.