Chapter 3

Question 1

Suppose U and V are subspaces of W. Then which of U⋃V and U∩V is a subspace?

Answer 1

U⋃V is not a subspace

U∩V is

Question 2

Let U and V be subspaces of some vector space W. In terms of a set, what is U + V?

Answer 2

U + V = {𝐮 + 𝐯: 𝐮 ⋲ U, 𝐯 ⋲ V}

Question 3

Is the sum of 2 subspaces a subspace?

Answer 3

Yes

Question 4

What does the sum of 2 subspaces contain?

Answer 4

U⋃V

Question 5

Suppose U and V are subspaces of some vector space. Then what is a direct sum?

Answer 5

U + V is called a direct sum, denoted U ⊕ V, if U ∩ V={𝟎}.

Question 6

If U, V ⊆ W with U = span({𝐮₁, …, 𝐮n}) and V = span({𝐯₁, …, 𝐯m}), then what is the span of W = U + V?

Answer 6

W = span({𝐮₁, …, 𝐮n, 𝐯₁, …, 𝐯m})

Question 7

If V = U ⊕ W, what does this mean about how every 𝐯 ⋲ V can be written?

Answer 7

Every 𝐯 ⋲ V can be written in a unique way as 𝐯 = 𝐮 + 𝐰 where 𝐮 ⋲ U and 𝐰 ⋲ W

Question 8

If U and W are subspaces of a finite-dimensional vector space, then what does this mean for dim(U ∩ W) and dim(U + V)?

Answer 8

dim(U + V) + dim(U ∩ W) = dim U + dim W

(In particular, if V = U ⊕ V, then dim V = dim U + dim W

Question 9

Let V be a vector space, and F = R or ℂ. What is an inner product?

Answer 9

A map V x V → F, (𝐮, 𝐯) →〈𝐮, 𝐯〉

Question 10

What does the bracket satisfy in the definition of the inner product?

Answer 10

(𝐮, 𝐯) →〈𝐮, 𝐯〉

1) conjugation symmetry: 〈𝐮, 𝐯〉= 〈𝐮, 𝐯〉(with line over top) for all 𝐮, 𝐯 ⋲ V
2) linearity: 〈𝐮 + 𝐮’, 𝐯〉= 〈𝐮, 𝐯〉+ 〈𝐮’, 𝐯〉and 〈𝛂𝐮, 𝐯〉= 𝛂〈𝐮, 𝐯〉
3) positive definite 〈𝐯, 𝐯〉>/= 0 and 〈𝐯, 𝐯〉= 0 ⤄ 𝐯 = 𝟎

Question 11

What is the standard inner product on R^n and ℂ^n?

Answer 11

〈𝐮, 𝐯〉= u₁v̅₁ + … + vnu̅n

Question 12

What is an inner product space?

Answer 12

A vector space over R or ℂ with an inner product

Question 13

In an inner product, what is the norm?

Answer 13

||𝐯|| = √〈𝐯, 𝐯〉

Question 14

What is another word for the norm in an inner product?

Answer 14

The length

Question 15

Why is the norm of an inner product real?

Answer 15

Due to positive definiteness, and is >/= 0

Question 16

In terms of inner product, when are 𝐯 and 𝐰 orthogonal?

Answer 16

If 〈𝐯, 𝐰〉= 0

Question 17

For the set of vectors {𝐯₁, …, vn}, when is it orthogonal?

Answer 17

This set is orthogonal if 〈𝐯i, 𝐯j〉= 0 for all i ≠ j

Question 18

For the set of vectors {𝐯₁, …, vn}, when is it orthonormal?

Answer 18

If ||𝐯i|| = 1 for all i

Question 19

When given an orthogonal set {𝐯₁, …, vn}, how do you make an orthonormal one?

Answer 19

{𝐯₁ / ||𝐯|| , …, vn / ||𝐯||}

Question 20

What is meant by normalising the set {𝐯₁, …, vn}?

Answer 20

Dividing each vector in the set by the norm

Question 21

What things do you need to show if you want to prove something is an inner product?

Answer 21

1) conjugation
2) linearity
3) positive definite

Question 22

What is the Gram-matrix of a set of vectors {𝐯₁, …, vn}?

Answer 22

The n by n matrix A with elements A_ij = 〈𝐯i, 𝐯j〉

try to write out diagram

Question 23

In terms of a Gram matrix, when are the vectors {𝐯₁, …, vn} linearly independent?

Answer 23

When the Gram matrix is non singular

Question 24

When is a matrix singular?

Answer 24

If A𝐮 = 𝟎 for some 𝐮

equivalent to detA = 0

Question 25

What is the Gram matrix of an orthogonal set?

Answer 25

A diagonal Gram matrix

Question 26

What is the Gram matrix of an orthonormal set?

Answer 26

The gram matrix is the identity matrix

Question 27

If 𝐯 ≠ 𝟎 and 𝐰 are vectors, then what is 𝐯 orthogonal to?

Answer 27

𝐰 - (⟨𝐰, 𝐯⟩/ ⟨𝐯, 𝐯⟩) 𝐯

Question 28

What is the Cauchy-Schwartz inequality?

Answer 28

Let 𝐮, 𝐯 belong to a inner product space. Then

|⟨𝐮, 𝐯⟩| <= ||𝐮|| ||𝐯||

Question 29

What is the triangle inequality?

Answer 29

For 𝐮, 𝐯 in an inner product space,

|| 𝐮 + 𝐯 || <= ||𝐮|| + ||𝐯||

Question 30

What is the Gram-Schmidt process?

Answer 30

Given a linearly independent set {𝐰₁, …, 𝐰n}, define a new set of vectors as

𝐯₁ = 𝐰₁

𝐯₂ = 𝐰₂ - (⟨𝐰₂, 𝐯₁⟩ / ⟨𝐯₁, 𝐯₁⟩) 𝐯₁

𝐯₃ = 𝐰₃ - (⟨𝐰₃, 𝐯₁⟩ / ⟨𝐯₁, 𝐯₁⟩) 𝐯₁ - (⟨𝐰₃, 𝐯₂⟩ / ⟨𝐯₂, 𝐯₂⟩) 𝐯₂

etc.

Question 31

What is true of the set{𝐯₁, …, 𝐯n} that is formed in the Gram-Schmidt process? How is an orthonormal set made from this?

Answer 31

{𝐯₁, …, 𝐯n} is an orthogonal set which is linearly independent and has the same span as {𝐰₁, …, 𝐰n}.

For an orthonormal set, normalise the 𝐯s.

Question 32

Does any finite dimensional inner product have an orthonormal basis?

Answer 32

Yes

Question 33

What is an upper triangular matrix?

Answer 33

Only the upper right hand side of the matrix is non-zero

i.e. R_ij = 0 for i > j

Question 34

If U is a subspace of V, what is the orthogonal subspace to U?

Answer 34

U⟂ = {𝐯 ⋲ V: ⟨𝐯, 𝐮⟩ = 0 ∀𝐮 ⋲ U}

Question 35

What is orthogonal decomposition?

Answer 35

If U is a finite-dimensional subspace of V, then V = U ⊕ U⟂

Question 36

Why is U⟂ important?

Answer 36

You can always divide finite-dimensional spaces into a subspace and its orthogonal subspace, due to orthogonal decomposition

Question 37

What is the best approximation theorem?

Answer 37

Let the vector space V = U ⊕ U⟂. Suppose that U has an orthogonal basis {𝐮₁, …, 𝐮n}. The element

𝐮 = (⟨𝐯, 𝐮₁⟩ / ⟨𝐮₁, 𝐮₁⟩) 𝐮₁ + …+ (⟨𝐯, 𝐮n⟩ / ⟨𝐮n, 𝐮n⟩) 𝐮n

is the unique element of U minimising ||𝐯 - 𝐮||

Question 38

How would you show U ⊕ V = R³?

Answer 38

1) show U+V = R³ using span

2) show U ∩ N = {𝟎}