Vector Subspaces

Goals: Know the test for a Linear Subspace. Be able to find the Nullspace of a matrix A. Understand the concept of linear combinations of vectors and spanning sets.

I. Vector Subspaces

In this section we begin to investigate the structure properties of Linear or Vector Spaces. One of the first structure properties to look for in a given vector space is what subspaces does it contain? In addition, we will see that it is possible to combine vector spaces together to form a new vector space and in this case the vector spaces that we join together become subspaces for the new vector space.

Our approach to subspaces is from the set-subset point of view. That is, given a subset S, (a set of vectors), of a vector space V, we want to be able to determine whether S is a subspace of V.

We begin with an example:

Let S = { ( x₁, x₂)^T| x₂ = 2x₁}. S is a subset of R².

What do vectors in S look like?

For two vectors u, v in S and a in R, show the operations of vector addition and scalar multiplication.

How can we tell if S is a subspace of R²?

We could check the eight axioms.

Nevertheless, since our subset lives inside of a vector space, you should realize that we don't need to check the all of the vector space axioms, but rather we only need to check:

Definition Subspace: If S is a nonempty subset of a vector space V, and S satisfies the following "closure" conditions:

i) au is an element of S whenever u is in S, for any scalar a.
ii) u + v is an element of S whenever u and v are elements of S.

then S is said to be a subspace of V.

This may be further simplified to:

Definition: Let S be a nonempty subset of a vector space V, then if for every u, v in S and scalar a, au + v Î S, S is a subspace of V.

II. Some Examples

Let's see how to apply this definition to test a set to see if it is a subspace.

Let S = {(x₁, x₂, x₃)^T | x₁ = x₂} and V = R³.

A)

Is S a subspace of R³?

B)

Let S = {(x, 1)^T| x is a real number}

What is V? Is S a subspace of V?

C)

Let S = { A e R^2x2 | a₁₂ = -a₂₁}

What is V? Is S a subspace of V?

D)

Let S = { P₃ = a₂x² +a₁x + a₀ | the constant term a₀ = 0}

What is V? Is S a subspace of V?

E)

Let S = { Cⁿ[a,b]}

S is the set of all continuous functions over the closed interval [a,b] that have n-th derivatives.

What is V? Is S a subspace?

F)

Let S = {0}

(the zero vector)

Prove: For any vector space V, S is a subspace of V.

G)

Let S = { 0000, 0100, 0010, 0110}

Let V = Z₄² is S a subsapce of V?

III. The Nullspace of a Matrix

Definition The Nullspace:
Let A be an mxn matrix. Let N(A) denote the set of all solutions to the homogeneous system Ax = 0. Thus,
N(A) = { x Î Rⁿ | A x = 0}.

Prove N(A) is a subspace of Rⁿ.

How to determine N(A) for a given A.

Determine here means describe the vectors in N(A).

Determine N(A) if:

Step 1) Using Gauss-Jordan reduction we put A into reduced row echelon form.

Step 2) Rewrite the associated equations.

The reduced form involves two free variables, and .

Thus,

Step 3) Set and .

The Nullspace of A consists of all vectors of the form:

where a and b are scalars.

Look at this last result, N(A) is composed of scalar multiples of the sum of the two vectors,

and

We say that N(A) is spanned by linear combinations of and .

IV. Spanning Sets

Linear combinations and spans are important enough to rate their own definition.

Definition Linear Combination: Let v₁, v₂, ..., v_n be vectors in a vector space V. A sum of the form a₁v₁ + a₂v₂ +... +a_nv_n, where the a_i's are scalars, is called a linear combination of v₁, v₂, ..., v_n.

Definition Span: The set of all linear combinations of v₁, v₂, ..., v_n is called
the span of v₁, v₂, ..., v_n. The span of v₁, v₂, ..., v_n. will be denoted by Span(v₁, v₂, ..., v_n).

The Unit Vectors e₁, e₂, e₃, ... , e_n in Rⁿ

The set of vectors of the form

are called the Unit Vectors of Rⁿ.

Find the span(e₁,e₂) in R³.

Prove: If v₁, v₂, ..., v_n are elements of a vector space V, then
the Span(v₁, v₂, ..., v_n) is a subspace of V.

Show that Span(0100, 0010) is { 0000, 0100, 0010, 0110}

Let S = { 0000, 0100, 0010, 0110}

If V = Z₄² , S is a subsapce of V.

Akamai: The importance of this theorem is that we can create subspaces of any given vector
space!

Definition Spanning Set: The set {v₁, v₂, ..., v_n} is a spanning set for V if and only if every vector in V can be written as a linear combination of v₁, v₂, ..., v_n.

Which of the following are spanning sets for R³?

a) {e₁, e₂, e₃, (1, 2, 3)^T}

b) {(1, 1, 1)^T , (0, 1, 0)^T}

Solution: To determine whether a set spans R³, we must determine if any arbitrary vector (a, b, c) in R³ can be written as a linear combination of the vectors in the given set.

For part a) can we find scalars

and

not all equal to zero, such that

Illustrate the span of two linearly independent vectors x and y in R³.

Let A be an nxn matrix. Prove that the following statements are equivalent.

a) N(A) = {O}

b) A is nonsingular

c) For each b in Rⁿ, the system Ax = b has a unique solution.

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Lecture 6 Notes