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Vector Spaces

Module by: Richard Baraniuk

Recall that we know how to add two signal/vectors and scale a vector/signal.

A vector space VV is a non empty subset of vectors such that given xx, y∈V y V , then x+y∈V x y V , and given x∈V x V , and a scalar αα, then αx∈V α x V . In other words, a vector space is closed under addition and scaling.

note:

Depending on whether α∈ℝ α or ℂ we can distinguish real vectors spaces from complex vector spaces.

Example 1: Vector Spaces

ℝ, ℝ2 2 , ℝ3 3 , ...... ℝN N

ℂ, ℂ2 2 , ℂ3 3 , ...... ℂN N

a line in ℝ2 2 through the origin

a line in ℝ3 3 through the origin

a plane in ℝ3 3 through the origin

...

0 0

Exercise 1

Vector spaces?

Figure 1
Figure 1 (spaces1.png)

A subspace of a vector space is a subset of the vectors that is itself a vector space.

Example 2

Vectors lying in a line through the origin in ℝ2 2 :

Figure 2
Figure 2 (spaces2.png)

Exercise 2

Fundamental idea in DSP: split the vector space of all signals into "signal space" and "noise space". Why is this reasonable?

Show that the set of all linear combinations of MM vectors in ℝN N is a vector space.

If a vector space consists of all linear combinations of MM vectors, then these vectors span the vector space. In other words, the spanning vectors can generate a vector in the vector space.

Example 3

The vector 11 1 1 spans this subspace in ℝ2 2 :

Figure 3
Figure 3 (spaces3.png)

Example 4

The canonical δδ basis, δ0...δN-1 δ 0 ... δ N 1 , span ℝN N .

Exercise 3

Can M<N M N vectors span ℝN N ?

Do M≥N M N vectors always span ℝN N ?

Exercise 4

Is ℝN N a subspace if ℂN N ?

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