Espaces vectoriels - partie 5 : sous-espace vectoriel (fin)

Understanding Vector Subspaces and Their Sums

Definition of Vector Subspace Sums

• The study of vector subspaces concludes with the definition of the sum of two vector subspaces, which leads to understanding supplementary vector subspaces. This will culminate in defining a generated subspace as the union of two subspaces.

Properties of Vector Subspace Sums

• The sum F + G of two vector subspaces F and G is not generally a vector subspace itself; thus, it is essential to identify the smallest vector space containing both F and G .

• The sum is defined as the set of all elements u + v , where u in F and v in G. This sum is denoted as F + G.

Characteristics of Vector Space Sums

• Two key properties highlight the significance of sums:

• First, F + G is indeed a vector subspace of a larger space E, unlike their union.

• Second, it represents the smallest vector space that contains both F and G. Understanding this requires examining specific examples.

Example in R³

• In an example where:

• Subspace F = (x,y,z) | y = 0, z = 0

• Subspace G = (x,y,z) | x = 0, z = 0

The element from the sum can be expressed as:

[ W = U + V,]

where each component belongs to its respective subspace. Thus, for this case, every element in the sum has coordinates such that they lie on a plane through the origin.

Unique Representation in R³

• It’s shown that every element in this example can be uniquely represented as a sum from each respective subspace:

[ W = (x,y,0)]

indicating that all elements form a plane at Z=0 within R³. However, some vectors may have multiple representations as sums from these spaces. For instance:

[ (1,2,3) = (0,2,3)+(1,0,3).]

This indicates non-uniqueness when combining elements from both spaces.

Direct Sum and Supplementary Spaces

Conditions for Direct Sum

• Two vector spaces are said to be in direct sum if:

• Their intersection only contains the zero vector:

• If F ∩ G = 0.

• Their combined span equals the entire space:

• If F + G = E.

If both conditions hold true for spaces within E then we denote them as supplementary spaces.

Importance of Supplementary Spaces

• Supplementary spaces provide unique decompositions; any element can be expressed uniquely as a combination from each space.

This means if an element can be written in two different ways using components from both spaces,

then those components must necessarily be equal:

[ W = U + V ⇒ U' + V' ⇒ U = U', V = V'.]

Examples and Verification

Checking Direct Sum Conditions

• To verify whether two given spaces are supplementary within R²:

• Check if their intersection consists solely of zero.

• Confirm that their combined span covers all possible vectors in R².

For instance:

• Given sets like:

• Set F: All points along X-axis.

• Set G: All points along Y-axis.

Both conditions are satisfied confirming they are supplementary within R² since any point can be represented uniquely by one point from each axis.

Further Examples with Distinct Lines

• In general terms,

Demonstrating the Sum of Subspaces in R³

Proving F + G is R³

• The goal is to show that the sum of subspaces F and G spans all of R³. A vector u with coordinates (x, y, z) must be expressed as a sum of vectors from F and G.

• Vectors V in F and W in G are defined such that u = V + W . Specifically, V = (y_1, z_1) and W = (x_2, 0, 0) .

• Through calculations, it is determined that y_1 = y , z_1 = z , and x_2 = x - y - z, confirming that any vector in R³ can be decomposed into elements from F and G.

Direct Sum of Vector Spaces

• The discussion shifts to the vector spaces P (even functions) and T (odd functions). It aims to prove that P and T form a direct sum within the space of functions from R to R.

• To find their intersection, if a function belongs to both P and T, it must be identically zero. This shows that the only function common to both spaces is the zero function.

Decomposition into Even and Odd Functions

• Any arbitrary function can be expressed as a sum of an even function (G defined by averaging f(x)) and an odd function (H defined by subtracting the average).

• The results confirm that every function in F can indeed be represented as a combination of functions from P and I. Thus, they are shown to be in direct sum.

Understanding Linear Combinations in Vector Spaces

Span Generated by Vectors

• For any finite set of vectors V_1, V_2,...V_n , their linear combinations form a subspace within E. This subspace is termed as being generated by these vectors.

• A vector belongs to this span if it can be expressed as a linear combination using scalars associated with each vector.

Properties of Generated Subspaces

• If another subspace contains all vectors from the generating set, then it must also contain their span. This highlights how generated subspaces relate to larger spaces.

Exploring Vector Spaces through Examples

One-Dimensional Subspaces

• The span created by a single non-zero vector U forms what’s known as a line through the origin in E.

Two-Dimensional Subspaces

• When considering two non-collinear vectors U and V in E, their span creates a plane within this space. An example illustrates this with specific coordinates for U and V.

Parametric Equations for Planes

• The relationship between X,Y,Z coordinates can be described parametrically based on λU + μV where λ & μ are real numbers. This leads to equations defining planes passing through origin containing U & V.

Function Spaces: Polynomial Representation

Polynomial Function Space Definition

• In discussing polynomial functions up to degree 2 defined by specific base functions like constant 1 or linear X, we establish how these generate polynomial spaces effectively capturing all quadratic forms.