M 340L Study Guide

8/2/2019 M 340L Study Guide

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M 340L Study Guide

4.1 Vector Spaces and Subspaces

Definition: A vector space is a nonempty set V of objects, called

vectors, on which are defined two operations, called addition andmultiplication by scalars (real numbers), subject to the ten axioms listed

below. The axioms must hold for all vetors u, v, and w in V and for all

scalars c and d.

(1)The sum ofu and v, denoted by u + v is in V.

(2) u + v = v + u

(3) (u + v) + w = u + (v + w)

(4)There is a zero vector 0 in V such that u + 0 = u.

(5) For each u in V, there is a vector u in V such that u +(-u) = 0

(6)The scalar multiple ofu by c, denoted by cu, is in V.

(7) c(u + v) = cu + cv.

(8) (c + d)u = cu + du

(9) c(du) = (cd)u.

(10) 1u = u.

SUBSPACES

Definition: A subspace of a vector space V is a subset H and Vthat

has three properties:

(a)The zero vector of V is in H2.

(b) H is closed under vector addition. That is, for each u and v in H,

the sum u + v is in H.

(c) H is closed under multiplication by scalars. That is, for each u inH and each scalar c, the vector cu is in H.

So every subspace is a vector space. Conversely, every vector

space is a subspace of itself and possibly other larger spaces.

The zero vector IS IN THE SUBSPACE OF V!!


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A SUBSPACE SPANNED BY A SET

As in Chapter 1, the term linear combination refers to any sum

of scalar multiples of vectors, and Span {v1,..,vp} denotes the

set og all vectors that can be written as linear combinations of

v1,..,vp.

This span is a plane.

Theorem 1: Ifv1,..,vp are in a vector space V, then Span { v1,..,vp}

is a subspace of V.

4.2 Null Spaces, Column Spaces, and Linear Transformations

THE NULL SPACE OF A MATRIX

Definition: The null space of and m x n matrix A, written as Nul A, isthe set of all solutions to the homogeneous equation Ax = 0. In set notation,

Nul A = {x : x is in Rn and Ax = 0}

A more dynamic description of Nul A is the set of all x in Rn that

are mapped into the zero vector of Rm via the linear

transformation x |-> Ax

Theorem 2: The null space of an m x n matrix A is a subspace of Rn.

Equivalently, the set of all solutions to a system Ax = 0 of m

homogeneous linear equations in n unknowns is a subspace of Rn.

It is important that the linear equations defining the set H are

homogeneous

Every linear combination ofu,v, and w is and element of Nul A.

Thus {u, v, w} is a spanning set for Nul A.

When Nul A contains nonzero vectors, the number of vectors in

the spanning set for Nul A equals the number of free variables

in the equation Ax = 0.THE COLUMN SPACE OF A MATRIX

Definition: The column space of an m x n matrix A, written as Col A, is

the set of all linear combinations of the columns of A. If A = [ a1.an ],

then


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Col A = Span { a1.an }

Theorem 3: The column space of an m x n matrix A is a subspace of Rm.

The notations Ax for vectors in Col A also shows that Col A is

the range of the linear transformation x |-> Ax.

The column space of an m x n matrix A is all of Rm if and only if

the equation Ax = b has a solution for each b in Rm.

4.3 Linearly Independent Sets; Bases

An indexed set of vectors {v1,., vp} in V is said to be linearly

indepenedent if the vector equation

c1v1 + c2v2 + . + cpvp = 0

has only the trivial solution.

Theorem 4: An indexed set {v1,, vp} of two or more vectors if and

only if some other vector besides the first is a linear combination of one

of the preceding vectors.

Definition: Let H be a subspace of a vector space V. An indexed set of

vectors B = {b1, ., bp} in V is a basis for H if

(i) B is a linearly independent set, and

(ii) the subspace spanned by B coincides with H; that is

H = Span {b1, ., bp}

THE SPANNING SET THEOREM

Theorem 5(The Spanning Set Theorem): Let S = {v1, ., vp} be a

set in V , and let H = Span {v1, ., vp}.


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(a) If one of the vectors in S is a linear combination of the

remaining vectors in S, then the set formed from S by

removing that vectors still spans H.

(b) If H doesnt = {0}, some subset of S is a basis for H.

BASES FOR NUL A AND COL A

Because the vectors that span the null space of a matrix

A, always produces a linearly independent set which is in

turn a basis for Nul A.

KEEP IN MIND that each nonpivot column of a matrix is alinear combination of the pivot columns in r.e.f.

Elementary row operations on a matrix do not affect the

linear dependence relationship among the columns of the

matrix.

Theorem 6: The pivot columns of a matrix A form a basis for Col A.

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M 340L Study Guide