MATH 337 Lecture #2

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PARALLELOGRAM RULE FOR ADDITION

• If u and v in R2 are represented as points in the plane, then u + v corresponds to the fourth vertex of the parallelogram whose other vertices are u, 0, and v.

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ALGEBRAIC PROPERTIES OF Rn

• The vector whose entries are all zero is called the zero vector and is denoted by 0.

• For all u, v, w in Rn and all scalars c and d:(i) (ii)(iii) (iv) ,

where denotes(v)(vi) (vii)(viii)

u v v u (u v) w u (v w) u 0 0 u u u ( u) u u 0

u ( 1)u(u v) u vc c c

( )u u uc d c d ( u)=(cd)(u)c d

1u u

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LINEAR COMBINATIONS

• Example 2: Let , and .

Determine whether b can be generated (or written) as a linear combination of a1 and a2. That is, determine whether weights x1 and x2 exist such that

(1)

If vector equation (1) has a solution, find it.

1

1

a 2

5

2

2

a 5

6

7

b 4

3

1 1 2 2a a bx x

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LINEAR COMBINATIONS

• A vector equation

has the same solution set as the linear system whose augmented matrix is

. (5)

• In particular, b can be generated by a linear combination of a1, …, an iff ∃ a solution to the linear system corresponding to the matrix (5).

1 1 2 2a a ... a b

n nx x x

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Spans

• Definition: If v1, …, vp are in Rn, then the set of all linear combinations of v1, …, vp is denoted by Span {v1, …, vp} and is called the subset of Rn spanned (or generated) by v1, …, vp. That is, Span {v1, ..., vp} is the collection of all vectors that can be written in the form

with c1, …, cp scalars.

1 1 2 2v v ... v

p pc c c

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Matrix-Vector Multiplication• Definition: If A is an mxn matrix, with columns a1, …,

an, and if x is in Rn, then the product of A and x, denoted by Ax, is the linear combination of the columns of A using the corresponding entries in x as weights; that is,

.

• Ax is defined only if the number of columns of Aequals the number of entries in x.

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MATRIX EQUATION Ax = b

• Theorem 3: If A is an mxn matrix, with columns a1, …, an, and if b is in Rn, then the matrix equation

has the same solution set as the vector equation

which, in turn, has the same solution set as the system of linear equations whose augmented matrix is

.

x bA

1 1 2 2a a ... b

n nx x x a

Existence

• For is Ax=b consistent for all b?A =

1 3 4

-4 2 -6

-3 -2 -7

é

ë

êêê

ù

û

úúú

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EXISTENCE OF SOLUTIONS• The equation has a solution iff b is a linear

combination of the columns of A.

• Theorem 4: Let A be an mxn matrix. Then the following statements are logically equivalent (i.e., either they are all true or they are all false).

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

b. Each b in Rm is a linear combination of the columns of A.

c. The columns of A span Rm.

d. A has a pivot position in every row.

x bA

Matrix-Vector Multiplication

• If the product Ax is defined, then the ith entry in Ax is the sum of the products of corresponding entries from row i of A and from the vector x.

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PROPERTIES OF THE MATRIX-VECTOR PRODUCT Ax

• Theorem 5: If A is an mxn matrix, u and v are vectors in Rn, and c is a scalar, then

a.

b. .

(u v) u v;A A A

( u) ( u)A c c A

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HOMOGENEOUS LINEAR SYSTEMS

• A system of linear equations is said to be homogeneous if it can be written in the form Ax=0, where A is an mxn matrix and 0 is the zero vector in Rn.

• Such a system Ax=b always has at least one solution, namely, x = 0 (the zero vector in Rn).

• This zero solution is usually called the trivial solution.

• The homogeneous equation Ax=0 has a nontrivial solution iff the equation has at least one free variable.

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HOMOGENEOUS LINEAR SYSTEMS

• Example 1: Determine if the following homogeneous system has a nontrivial solution. Then describe the solution set.

1 2 3

1 2 3

1 2 3

3 5 4 0

3 2 4 0

6 8 0

x x x

x x x

x x x

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HOMOGENEOUS LINEAR SYSTEMS

• Example 2: Determine if the following homogeneous system has a nontrivial solution. Then describe the solution set.

10x1 – 3x2 – 2x3 = 0

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SOLUTIONS OF NONHOMOGENEOUS SYSTEMS

• When a nonhomogeneous linear system has many solutions, the general solution can be written in parametric vector form as one vector plus an arbitrary linear combination of vectors that satisfy the corresponding homogeneous system.

• Example 2: Describe all solutions of Ax=b, where3 5 4

3 2 4

6 1 8

A

7

b 1

4

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SOLUTIONS OF NONHOMOGENEOUS SYSTEMS

• Theorem 6: Suppose the equation Ax=b is consistent for some given b, and let p be a solution. Then the solution set of Ax=b is the set of all vectors of the form w=p+vh, where vh is any solution of the homogeneous equation Ax=0.

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WRITING A SOLUTION SET (OF A CONSISTENT SYSTEM) IN PARAMETRIC

VECTOR FORM1. Row reduce the augmented matrix to reduced

echelon form.

2. Express each basic variable in terms of any free variables appearing in an equation.

3. Write a typical solution x as a vector whose entries depend on the free variables, if any.

4. Decompose x into a linear combination of vectors (with numeric entries) using the free variables as parameters.