Chapter 4: Real vector spaces

Question 1

Vector space

Answer 1

A set of objects that are closed under addition and scalar multiplication.

Question 2

Closure under addition

Answer 2

If u and v are objects in V, then u + v is in V.

Question 3

Closure under scalar multiplication

Answer 3

if k is any scalar and u is any object in V, then ku is in V.

Question 4

Subspace

Answer 4

A subset of a vector space (say V), that is itself a vector space under the addition and scalar multiplication defined on V.

Question 5

If W is a set of one or more vectors in a vector space V, then W is a subspace of V iff:

Answer 5

• u and v are vector is W, then u+v is in W.
• k is any scalar and u is any vector in W, then ku is in W.

Question 6

Create a new subspace from known subspaces:

Answer 6

If W₁, W₂, … W_r are subspaces of a vector space V, then the intersection of these subspaces is also a subspace of V.

Question 7

Span of S

Answer 7

The subspace of a vector space V, that is formed from all possible linear combinations of the vectors in a nonempty set S.

We say that S span that subspace.

Question 8

A subspace of Rⁿ using the matrix A.

Answer 8

The solution set of Ax = 0, in n unknowns.

Question 9

Theorem 4.2.5, considering the equality of the spans of 2 vector spaces.

Answer 9

If S = {v₁, v₂, …, v_r} and S’ = {w₁, w₂, …, w_k} are nonempty sets of vectors in a vector space V, then span{v₁, v₂, …, v_r} = span{w₁, w₂, …, w_k} iff each vector in S is a linear combination of those in S’, and each vector in S’ is a linear combination of those in S.

Question 10

Definition of linear independence:

Answer 10

If S = {v₁, v₂, …, v_r} is a nonempty set of vectors in a vector space V, then the vector equation k₁v₁ + k₂v₂ + … + k_rv_r = 0 has at least one solution, namely, k₁ = k₂ = … = k_r = 0. We call this the trivial solution. If this is the only solution, then S is said to be a linearly independent set.

Question 11

Interpretation of linear independence:

Answer 11

A set with 2+ vectors is: Linearly independent iff no vector in S is expressible as a linear combination of the other vectors in S.

Question 12

Basic theorem concerned with linear independence ( 1 or 2 vectors):

Answer 12

• A finite set that contains 0 is linearly dependent.
• A set with exactly one vector is linearly independent iff that vector is not 0.
• A set with exactly 2 vectors is linearly independent iff neither vector is a scalar multiple of the other.

Question 13

Let S = {v₁, v₂, …, v_r} be a set of vectors in Rⁿ. When is the set linearly dependent?

Answer 13

If r > n

Question 14

Wronskian of f₁, f₂, …, f_n

Answer 14

If f₁ = f₁ (x), f₂ = f₂ (x), …, f_{n<strong> </strong>}= f_n (x) are functions that are n-1 times differentiable on the interval (-∞, ∞):

Question 15

Theorem 4.3.4 using the Wronskian to determine linear independence of functions

Answer 15

If the functions f₁, f₂,…, f_n have n-1 continous derivatives on the interval (-∞, ∞)

and if the Wronskian of these functions is not identically zero on (-∞, ∞), then these functions form a linearly independent set of vectors in C^(n-1)(-∞, ∞)

Question 16

Define a Basis

Answer 16

If V is any vector space and S = {v₁, v₂, …, v_n} is a finite set of vectors in V, then S is called a basis for V if the following 2 conditions hold:

• S is linearly independent
• S spans V.

Question 17

Theorem 4.4.1: Uniqueness of Basis Representation

Answer 17

If S = {v₁, v₂, … v_n} is a basis for a vector space V, then every vector v in V can be expressed in the form v = c₁v₁ + c₂v₂ + … + c_nv_n in exactly one way.

Question 18

Definition for a coordinate vector of v relative to S.

Answer 18

If S = {v₁, v₂,… v_n} is a basis for a vector space V, and v = c₁v₁+ c₂v₂ + … + c_nv_n

is the expression for a vector v in terms of the basis S, then the scalars c₁, c₂, … c_n are called the coordinates of v relative to the basis S. The vector (c₁, c₂, … c_n) in Rⁿ constructed from these coordinates is called the coordinate vector of v relative to S*; * it is denoted by

(v)_S = (c₁, c₂, … c_n)

Question 19

Theorem 4.5.1 on the number of vectors in a basis.

Answer 19

All bases for a finite-dimensional vector space have the same number of vectors.

Question 20

Theorem 4.5.2 on the a basis of a vector space and linear independence

Answer 20

Let V be a finite dimensional vector space, and let {v₁, v₂,… v_n} be any basis.

• If a set has more than n vectors, then it is linearly dependent.
• If a set has fewer than n vectors, then it does not span V.

Question 21

Dimension of a finite-dimensional vector space V.

Answer 21

The number of vectors in a basis for V.

The zero vector space has a dimension of zero.

Question 22

Dimension of 0

Answer 22

0

Question 23

Plus/Minus Theorem

Answer 23

Let S be a nonempty set of vectors in a vector space V.

• If S is a linearly independent set and if v is a vector in V that is outside of span(S), then the set S ⋃ {v} that results by inserting v into S is still linearly independent.
• If v is a vector in S that is expressible as a linear combination of other vectors in S, and if S - {v} denotes the set obtained by removing v from S, then S and S - {v} span the same space; that is,

span(S) = span(S - {v})

Question 24

Theorem 4.5.4 on the conditions of a basis.

Answer 24

Let V be an n-dimensional vector space, and let S be a set in V with exactly n vectors.

Then S is a basis for V if and only if S spans V or S is linearly independent.

Question 25

Theorem 4.5.6 that relates the dimension of a vector space to the dimensions of its subspaces.

Answer 25

If W is a subspace of a finite-dimensional vector space V, then:

• W is finite-dimensional
• dim(W) ≤ dim(V)
• W = V if and only if dim(W) = dim(V).

Question 26

The columns of a transition matrix

Answer 26

Are the coordinate vectors of the old basis, relative to the new basis.

Question 27

Theorem 4.6.1 on the invertibility of transition matrices

Answer 27

If P is the transition matrix from a basis B’ to a basis B for a finite-dimensional vector space V, then P is invertible

and P^-1 is the transition matrix from B to B’.

Question 28

The procedure for computing P_{B → B’}

Answer 28

Form the matrix [B’ | B]

Use elementary row operations to reduce the matrix to reduced row echelon form.

The resulting matrix will be [I | P_{B → B’}]

[new basis | old basis] → [I | transition from old to new]

Question 29

Theorem 4.6.2 on the transitionto the Standard Basis

Answer 29

Let B’ = {u₁, u₂, … , u_n} be any basis for the vector space Rⁿ and let S = {e₁, e₂, … , e_n} be the standard basis for Rⁿ. If the vectors in these basis are written in column form, then:

P_B’→S = [u₁ | u₂ | … | u_n]

Question 30

Row space of A.

Answer 30

The subspace spanned by the row vectors of A.

Question 31

Column space of A

Answer 31

The subspace spanned by the column vectors of A.

Question 32

Null space of A

Answer 32

Solution space of the homogeneous system of equations Ax = 0.

Question 33

Theorem 4.7.2 on the null space and elementary row operations

Answer 33

Elementary row operations do not change the null space of a matrix

Question 34

Theorem 4.7.4 on the row space and elementary row operations

Answer 34

Elementary row operations do not change the row space of a matrix.

Question 35

Theorem 4.8.1 on the dimesnion of the row and column space of a matrix

Answer 35

The row space and column space of a matrix A have the same dimension.

Question 36

Rank of matrix A

Answer 36

The common dimension of the row space and column space of A.

Question 37

Nullity of A

Answer 37

The dimension of the null space of A.

Question 38

Theorem 4.8.2: Dimension Theorem for Matrices

Answer 38

If matrix A has n columns:

rank(A) + nullity(A) = n

Question 39

Theorem 4.8.3 on rank & nullity

Answer 39

If A is an m x n matrix, then:

• rank(A) = the number of leading variables in the general solution of Ax = 0
• nullity(A) = the number of parameters in the general solution of Ax = 0

Question 40

Number of parameters contained in the general solution of a consistent linear system Ax = b​,

of m equations in n unknowns, where

A has rank r

Answer 40

n - r parameters.

Question 41

Orthogonal complement of W

Answer 41

The set of all vectors that are orthogonal to every vector in W.

Question 42

Theorem 4.8.9 on orthogonal complements & null, row, column space

Answer 42

If A is an m x n matrix:

• The null space of A and the row space of A are orthogonal complements.
• The null space of A^T and the column space of A are orthogonal complements.

Question 43

Theorem 4.9.1

Properties of matrix transfrmations:

Answer 43

• T_A(0) = 0
• T_A(ku) = kT_A(u)
• T_A(u + v) = T_A(u) + T_A(v)
• T_A(u - v) = T_A(u) - T_A(v)

Question 44

Steps:

Finding the Standard Matrix for a Matrix Transformation

Answer 44

• Find the images of the standard basis vectors e₁, e₂, … e_n for Rⁿ in column form.
• Construct the matrix that has the images obtained in Step 1 as its successive columns.

Question 45

4.10.1: Equivalent statements for transformation matrix A:

Answer 45

• A is invertible
• The range of T_A is Rⁿ
• T_A is one-to-one

Question 46

4.10.2: Relationships between u, v and matrix Transformations

Answer 46

• T(u + v) = T(u) + T(v)
• T(k u) = k T(u)