Chapter 3 Flashcards
Suppose U and V are subspaces of W. Then which of U⋃V and U∩V is a subspace?
U⋃V is not a subspace
U∩V is
Let U and V be subspaces of some vector space W. In terms of a set, what is U + V?
U + V = {𝐮 + 𝐯: 𝐮 ⋲ U, 𝐯 ⋲ V}
Is the sum of 2 subspaces a subspace?
Yes
What does the sum of 2 subspaces contain?
U⋃V
Suppose U and V are subspaces of some vector space. Then what is a direct sum?
U + V is called a direct sum, denoted U ⊕ V, if U ∩ V={𝟎}.
If U, V ⊆ W with U = span({𝐮₁, …, 𝐮n}) and V = span({𝐯₁, …, 𝐯m}), then what is the span of W = U + V?
W = span({𝐮₁, …, 𝐮n, 𝐯₁, …, 𝐯m})
If V = U ⊕ W, what does this mean about how every 𝐯 ⋲ V can be written?
Every 𝐯 ⋲ V can be written in a unique way as 𝐯 = 𝐮 + 𝐰 where 𝐮 ⋲ U and 𝐰 ⋲ W
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)?
dim(U + V) + dim(U ∩ W) = dim U + dim W
(In particular, if V = U ⊕ V, then dim V = dim U + dim W
Let V be a vector space, and F = R or ℂ. What is an inner product?
A map V x V → F, (𝐮, 𝐯) →〈𝐮, 𝐯〉
What does the bracket satisfy in the definition of the inner product?
(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
What is the standard inner product on R^n and ℂ^n?
〈𝐮, 𝐯〉= u₁v̅₁ + … + vnu̅n
What is an inner product space?
A vector space over R or ℂ with an inner product
In an inner product, what is the norm?
||𝐯|| = √〈𝐯, 𝐯〉
What is another word for the norm in an inner product?
The length
Why is the norm of an inner product real?
Due to positive definiteness, and is >/= 0
In terms of inner product, when are 𝐯 and 𝐰 orthogonal?
If 〈𝐯, 𝐰〉= 0
For the set of vectors {𝐯₁, …, vn}, when is it orthogonal?
This set is orthogonal if 〈𝐯i, 𝐯j〉= 0 for all i ≠ j
For the set of vectors {𝐯₁, …, vn}, when is it orthonormal?
If ||𝐯i|| = 1 for all i
When given an orthogonal set {𝐯₁, …, vn}, how do you make an orthonormal one?
{𝐯₁ / ||𝐯|| , …, vn / ||𝐯||}
What is meant by normalising the set {𝐯₁, …, vn}?
Dividing each vector in the set by the norm
What things do you need to show if you want to prove something is an inner product?
1) conjugation
2) linearity
3) positive definite
What is the Gram-matrix of a set of vectors {𝐯₁, …, vn}?
The n by n matrix A with elements A_ij = 〈𝐯i, 𝐯j〉
try to write out diagram
In terms of a Gram matrix, when are the vectors {𝐯₁, …, vn} linearly independent?
When the Gram matrix is non singular
When is a matrix singular?
If A𝐮 = 𝟎 for some 𝐮
equivalent to detA = 0