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Li Bao bin | UCAS 9 / 56
Vector Spaces | Vector Spaces and Subspaces | Subspaces
Example 8
For a set of vectors S = {a1 , a2 , · · · , an } from a subspace V ⊆ Rm×1 , let A be the matrix containing the ai ’s as its columns. S spans V if and only if for each b ∈ V , there corresponds a column x such that Ax = b. This simple observation often is quite helpful. For example, to test whether or not S = {(1, 1, 1), (1, −1, −1), (3, 1, 1)} spans R3 .
Example 3
With function addition and scalar multiplication defined by (f + g )(x) = f (x) + g (x) and (αf )(x) = αf (x), the following sets are vector spaces over R: 1. The set of functions mapping the interval [0, 1] into R. 2. The set of all real-valued continuous functions defined on [0, 1]. 3. The set of real-valued functions that are differentiable on [0, 1]. 4. The set of all polynomials with real coefficients.
Vector Spaces | Vector Spaces and Subspaces | Subspaces
Example 4
Given a vector space V , the set Z = 0 containing only the zero vector is a subspace of V . This subspace is called the trivial subspace.
Li Bao bin | Spaces | Vector Spaces and Subspaces | Vector Spaces
Li Bao bin | UCAS
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Vector Spaces | Vector Spaces and Subspaces | Vector Spaces
In R3 , Planes through the origin are also subspaces.
Questions:
x What about straight lines not through the origin? x What about curved lines through the origin?
Example 1
The set Rm×n of m × n real matrices is a vector space over R. The set C m×n of m × n real matrices is a vector space over C .
Li Bao bin | UCAS 4 / 56
The formal definition of a vector space stipulates how these four things relate to each other. V is a nonempty set of objects called vectors. Although V can be quite general, we will usually consider V to be a set of n-tuples or a set of matrices. F is a scalar field)for us F is either the field R of real numbers or the field C of complex numbers. Vector addition (denoted by x + y ) is an operation between elements of V. Scalar multiplication (denoted by αx ) is an operation between elements of F and V .
Vector Spaces | Vector Spaces and Subspaces | Vector Spaces
Example 2
The real coordinate spaces R1×n = {(x1 , x2 , · · · , xn ), xi ∈ R} Rn×1 = {(x1 , x2 , · · · , xn )T , xi ∈ R}
Li Bao bin | UCAS
8 / 56
Vector Spaces | Vector Spaces and Subspaces | Subspaces
Example 7
1.The unit vectors {e1 , e2 , · · · , en } form a spanning set for Rn . 2.The finite set {1, x, x2 , · · · , xn } spans the space of all polynomials such that deg p(x) ≤ n, and the infinite set {1, x, x2 , · · · } spans the space of all polynomials.
Li Bao bin | UCAS
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Vector Spaces | Vector Spaces and Subspaces | Subspaces
Example 5
Straight lines through the origin in R2 and R3 are subspaces.
Example 6
Li Bao bin | UCAS 10 / 56
Vector Spaces | Vector Spaces and Subspaces | Subspaces
The sum X + Y is also a subspace of V . If SX , SY span X , Y , then SX SY spans X + Y .
Li Bao bin | UCAS
11 / 56
Vector Spaces
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Vector Spaces
Baobin Li Email:libb@
School of Computer and Control Engineering, UCAS
Li Bao bin | UCAS
1 / 56
Vector Spaces | Vector Spaces and Subspaces | Vector Spaces
Just place these row as columns in a matrix A. Check ”Is the system Ax = b consistent for every b ∈ R3 ? 1 1 3 x1 b1 1 −1 1 x2 = b2 1 −1 1 x3 b3 As we know, Ax = b is consistent if and only if rank [A|b] = rank (A). In this case, rank (A) = 2, but rank [A|b] = 3 for some b (e.g., b1 = 0, b2 = 1, b3 = 0), so S doesn’t span R3 .
Example 9
If X ⊆ R2 and Y ⊆ R2 are subspaces defined by two different lines through the origin, then X + Y = R2 . This follows from the parallelogram law.
Spaces and Subspaces
Many mathematical entities that were considered to be quite different from matrices were in fact quite similar. For example, objects such as points in the plane R2 and R3 , polynomials, continuous functions, and differentiable functions satisfy the same additive properties and scalar multiplication properties given for matrices. Rather than studying each topic separately, it is more efficient and productive to study many topics at one time by studying the common properties that they satisfy. This eventually led to the axiomatic definition of a vector space. A vector space involves four things)two sets V and F , and two algebraic operations called vector addition and scalar multiplication.
