Chapter 4 – Key Concepts

Question 1

What is a vector space?

Answer 1

A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers), subject to the ten axioms (or rules) listed below. The axioms must hold for all vectors u, v, and w in V and for all scalars c and d:

1. u + v ∈ V.
2. u + v = v + u
3. (u + v) + w = u + (v + w)
4. ∃ 0 ∈ V s.t. v + 0 = v
5. ∀ u ∈ V, ∃ (-u) ∈ V s.t. u + (-u) = 0
6. The scalar multiple of u by c, denoted by cu, is in V.
7. c(u + v) = cu + cv
8. (c + d)u = cu + du
9. c(du) = (cd)u
10. 1u = u

Question 2

For each u in a vector space V and scalar c, evaluate the following:

(1) 0u
(2) c0
(3) -u

Answer 2

(1) 0u = 0
(2) c0 = 0
(3) -u = (-1)u

Question 3

What does it mean for H to be a subspace of a vector space V?

Answer 3

A subspace of a vector space V is a subset H of V that has three properties:

a. 0 ∈ H
b. ∀ u, v ∈ H, u + v ∈ H
c. ∀ u ∈ H and c ∈ ℝ, cu ∈ H

Question 4

If v₁, …, v_p are in a vector space V, what must be true about Span{v₁, …, v_p}?

Answer 4

Span{v₁, …, v_p} is a subspace of V.

Question 5

What is the null space of an m×n matrix A?

Answer 5

The null space of an m×n matrix A, written as Nul A, is the set of all solutions of the homogeneous equation Ax = 0. In set notation, Nul A = {x : x ∈ ℝⁿ and Ax = 0}

Question 6

The null space of an m×n A is a subspace of what? In other words, the set of all solutions to Ax = 0 is a subspace of what?

Answer 6

ℝⁿ

Question 7

What is the column space of an n×m matrix A?

Answer 7

The column space of an m×n matrix A, written as Col A, is the set of all linear combinations of the columns of A. If A = [a₁ … a_n], then Col A = Span{a₁, …, a_n}.

Question 8

The column space of an m×n matrix A is is a subspace of what?

Answer 8

ℝ^m

Question 9

For A is m×n, describe Col A in set notation.

Answer 9

Col A = {b : b = Ax for some x in ℝⁿ}

Question 10

If the column space of an m×n matrix A is all of ℝ^m, what must be true about the equation Ax = b?

Answer 10

The column space of an m×n matrix A is all of ℝ^m if and only if the equation Ax = b has a solution for all b ∈ ℝ^m.

Question 11

What is the definition of a linear transformation from a vector space V into a vector space W?

Answer 11

A linear transformation from a vector space V into a vector space W is a rule that assigns to each vector x in V a unique vector T(x) in W s.t.:

(i) T(u + v) = T(u) + T(v) ∀ u, v ∈ V, and
(ii) T(cu) = cT(u) ∀ u ∈ V and c ∈ ℝ.

Question 12

If an indexed set {v₁, …, v_p} of two or more vectors is linearly dependent and v₁ ≠ 0, what must be true about an arbitrary vector v_j for j > 1, in relation to v₁, …, v_j-1?

Answer 12

There must be some v_j that is a linear combination of v₁, …, v_j-1.

Question 13

If H is a subspace of V, what must be true about B = {b₁, …, b_p} in V for it to be a basis for H?

Answer 13

(i) B is a linearly independent set, and
(ii) the subspace spanned be B coincides with H. That is, H = span{b₁, …, b_p}.

Question 14

What does the spanning set theorem say?

Answer 14

Let S = {v₁, …, v_p} be a set in V, and let H = span{v₁, …, v_p}.

a. If v_k ∈ S is a linear combination of the remaining vectors in S, then S \ {v_k}.
b. If H ≠ {0}, some subset of S is a basis for H.

Question 15

Which columns of a matrix A form the basis for Col(A)?

Answer 15

The pivot columns of a matrix A form a basis for Col A.

Question 16

What does the unique representation theorem say?

Answer 16

Let B = {b₁, …, b_n} be a basis for a vector space V. Then ∀ x ∈ V, ∃ a unique set of scalars c₁, …, c_n such that: x = c₁b₁ + … + c_nb_n.

Question 17

Suppose B = {b₁, …, b_n} is a basis for V, and x ∈ V. What are the coordinates of x relative to the basis B (or the B-coordinates of x)?

Answer 17

the coordinates of x relative to the basis B (or the B-coordinates of x) are the weights c₁, …, c_n such that x = c₁b₁ + … + c_nb_n.

Question 18

What is the change of coordinate matrix for a basis B?

Answer 18

P_B = [b₁ … b_n] s.t. x = P_B[x]_B.

Question 19

Let B = {b₁, …, b_n} be a basis for V. Is the coordinate mapping x ↦ [x]_B one-to-one or onto?

Answer 19

x ↦ [x]_B is a one-to-one linear transformation from V onto ℝⁿ.

Question 20

If a vector space V has a basis B = {b₁, …, b_n}, then is a set of more than n vectors linearly independent or dependent?

Answer 20

It must be linearly dependent.

Question 21

If a vector space V has a basis with n vectors, what must be true about the number of vectors in any other basis for V?

Answer 21

They must all contain exactly n vectors as well.

Question 22

What does it mean for a vector space V to be finite-dimensional? Infinite-dimensional? What’s the dimension of {0}?

Answer 22

If V is spanned by a finite set, then V is said to be finite-dimensional, and the dimension of V , written as dim V , is the number of vectors in a basis for V . The dimension of the zero vector space {0} is defined to be zero. If V is not spanned by a finite set, then V is infinite-dimensional.

Question 23

Let H be a subspace of a finite-dimensional vector space V. What kinds of subsets of H and V can we expand to a basis for H? Is H finite-dimensional? What is the relationship between dim H and dim V?

Answer 23

Any linearly independent set in H can be expanded to a basis for H. H is finite-dimensional and dim H ≤ dim V.

Question 24

What is the basis theorem?

Answer 24

Let V be a p-dimensional vector space, p ≥ 1. Any linearly independent set of exactly p elements in V is automatically a basis for V. Any set of exactly p elements that spans V is automatically a basis for V.

Question 25

What is the relationship between the dimensions of the null space of A and the column space of A and the solution set free variables and pivot columns in A?

Answer 25

The dimension of Nul A is the number of free variables in the equation Ax = 0, and the dimension of Col A is the number of pivot columns in A.

Question 26

What is the relationship between the row spaces of two row equivalent matrices A and B? How can bases for A and B be generated by elementary row operations?

Answer 26

If two matrices A and B are row equivalent, then their row spaces are the same. If B is in echelon form, the nonzero rows of B form a basis for the row space of A as well as for that of B.

Question 27

What is the rank of a matrix A?

Answer 27

The rank of A is the dimension of the column space of A.

Question 28

What is the rank theorem?

Answer 28

The dimensions of the column space and the row space of an m×n matrix A are equal. This common dimension, the rank of A, also equals the number of pivot positions in A and satisfies the equation:

rank A + dim Nul A = n

Question 29

Evaluate the following equivalent expressions for the continued invertible matrix theorem w/ rank. A is an n×n matrix, the following are equivalent to the statement that A is invertible:

m. The columns of A form a basis for what?
n. Col A = ?
o. dim Col A = ?
p. rank A = ?
q. Nul A = ?
r. dim Nul A = ?

Answer 29

m. The columns of A form a basis of ℝⁿ.
n. Col A = ℝⁿ
o. dim Col A = n
p. rank A = n
q. Nul A = {0}
r. dim Nul A = 0

Question 30

Let B = {b₁, …, b_n} and C = {c₁, …, c_n} be bases of a vector space V. What is the change-of-coordinates matrix from B to C?

Answer 30

An n×n matrix P(C←B) s.t. [x]_C = P(C←B) [x]_B. The columns of P(C←B) are the C-coordinate vectors of the vectors in B: P(C←B) = [[b₁]_C … [b_n]_C ]

Question 31

What is the relationship between P(C←B) and P(B←C)?

Answer 31

( P(C←B) )^-1 = P(B←C)