lecture 3 section 1-7, 1-8 and 1-9
TRANSCRIPT
1
1.7
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Linear Equationsin Linear Algebra
LINEAR INDEPENDENCE
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LINEAR INDEPENDENCE - Definition
Definition: set of vectors {v1, …, vp} in Rn is said to be linearly independent if the vector equation
x1v1 + x1v1 + …+ x1v1 = 0
has only the trivial solution. Otherwise: linearly dependent
Nontrivial solution exists linear dependence relation
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LINEAR INDEPENDENCE - Example
Example 1: Let , , and .
a. Is the set {v1, v2, v3} linearly independent?
b. If possible, find a linear dependence relation among v1, v2, and v3.
1
1
v 2
3
=
2
4
v 5
6
=
3
2
v 1
0
=
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LINEAR INDEPENDENCE OF MATRIX COLUMNS matrix A = [a1 a2 … a3]
The matrix equation Ax = 0 can be written asx1v1 + x1v1 + …+ x1v1 = 0
Each linear dependence relation among the columns of A corresponds to a nontrivial solution of Ax = 0.
Thus, the columns of matrix A are linearly independent iff the equation Ax = 0 has only the trivial solution.
Matrix Example
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SETS OF ONE VECTOR
A set containing only one vector – say, v – is linearly independent iff v is not the zero vector.
This is because the vector equation has only the trivial solution when .
The zero vector is linearly dependent because has many nontrivial solutions.
1v 0x =v 0≠
10 0x =
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SETS OF TWO VECTORS
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Characterization of Linearly Dependent Sets
Theorem 7: An indexed set S = {v1, …, vp} of vectors is linearly dependent iff at least one of the vectors in S is a linear combination of the others.
In fact, if S is linearly dependent and j > 1, then some vj (with v1 ≠ 0) is a linear combination of the preceding vectors, v1, …, vj-1.
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SETS OF TWO OR MORE VECTORS
Proof: If some vj in S equals a linear combination of the other vectors, then vj can be subtracted from both sides of the equation, producing a linear dependence relation with a nonzero weight on vj.
v1 = c2 v2 + c3 v3
0 = (-1)v1 + c2 v2 + c3 v3 + 0v4 + … 0vp
Thus S is linearly dependent. Conversely, suppose S is linearly dependent.
If v1 is zero, then it is a (trivial) linear combination of the other vectors in S.
( 1)−
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SETS OF TWO OR MORE VECTORS
Otherwise, , and there exist weights c1, …, cp, not all zero, such that
.
Let j be the largest subscript for which .
If , then , which is impossible because
.
1v 0≠
1 1 2 2v v ... v 0p pc c c+ + + =
0jc ≠
1j = 1 1v 0c =1v 0≠
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SETS OF TWO OR MORE VECTORS
So , and1j >
1 1 1v ... v 0v 0v ... 0v 0j j j j pc c ++ + + + + + =
1 1 1 1v v ... vj j j jc c c − −= − − −
111 1v v ... v .j
j j
j j
cc
c c−
−
= − + + − ÷ ÷
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SETS OF TWO OR MORE VECTORS Theorem 7 does not say that every vector in a linearly
dependent set is a linear combination of the preceding vectors.
A vector in a linearly dependent set may fail to be a linear combination of the other vectors.
Example 2: Let and . Describe the
set spanned by u and v, and explain why a vector w is in Span {u, v} if and only if {u, v, w} is linearly dependent.
3
u 1
0
=
1
v 6
0
=
Example
Example 2: Let and .
a)Describe the set spanned by u and v b)explain why a vector w is in Span {u, v} iff
{u, v, w} is linearly dependent.
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3
u 1
0
=
1
v 6
0
=
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SETS OF TWO OR MORE VECTORS So w is in Span {u, v}. See the figures given below.
Example 2 generalizes to any set {u, v, w} in R3 with u and v linearly independent.
The set {u, v, w} will be linearly dependent iff w is in the plane spanned by u and v.
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SETS OF TWO OR MORE VECTORS
Theorem 8: If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. That is, any set {v1, …, vp} in Rn is linearly dependent if .
Proof: Let . Then A is , and the equation Ax = 0
corresponds to a system of n equations in p unknowns.
If , more variables than equations, so there must be a free variable many solutions
p n>1v v pA = L
n p×
p n>
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SETS OF TWO OR MORE VECTORS
Theorem 9: If a set in Rn contains the zero vector, then the set is linearly dependent.
Proof: By renumbering the vectors, we may suppose v1 = 0
1v1 + 0v2 + … + 0vp = 0 Nontrivial solution linearly dependent
1{v ,..., v }pS =
Linear Transformations
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A x = b
A u = 0
Linear Transformations
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Domain (R4) Co-Domain (R2)
x
T(x)
b
b
T(x)
0
Range
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LINEAR TRANSFORMATIONS A transformation (or function or mapping) T from Rn
to Rm is a rule that assigns to each vector x in Rn a vector T (x) in Rm
Rn is called domain of T Rm is called the codomain of T. Notation T: Rn Rm For x in Rn, the vector T (x) in Rm is called the image of
x (under the action of T ). The set of all images T (x) is called the range of T.
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MATRIX TRANSFORMATIONS
For each x in Rn, T (x) is computed as Ax, where A is an mxn matrix.
Notation for a matrix transformation by x Ax Domain of T is Rn when A has n columns Co-domain of T is Rm when A has m rows The range of T is the set of all linear combinations of
the columns of A, because each image T (x) is of the form Ax.
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MATRIX TRANSFORMATIONS - ExampleExample 1: Let T: Rn Rm by T(x)=Ax
a.Find T (u), the image of u under the transformation T.
b.Find an x in R2 whose image under T is b.
c.Is there more than one x whose image under T is b?
d.Determine if c is in the range of the transformation T.
1 3
3 5
1 7
A
− = −
2u
1
= −
3
c 2
5
= −
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LINEAR TRANSFORMATIONS
Definition: A transformation (or mapping) T is linear if:
i. for all u, v in the domain of T:T(u + v) = T(u) + T(v)
ii. for all scalars c and all u in the domain of T:
T(cu) = cT(u) Linear transformations preserve the operations of
vector addition and scalar multiplication.
Linear Transformations
These two properties lead to the following useful facts.If T is a linear transformation, then:
T(0) = 0T(c1v1 + c2v2 +…+ cpvp) = c1T(v1) + c2T(v2) +…+ cpT( vp)
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SHEAR TRANSFORMATION
Example 2: Let . The transformation
defined by T = Ax is called a shear transformation.
1 3
0 1A
=
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Contraction/Dilation
Given a scalar r, define T : Rn Rn by T(x) = rx
contraction when 0 ≤ r < 1 dilation when r > 1
How to Find Matrix of Transformation
Theorem 10: T: Rn Rm is linear transformation, then a unique matrix A such that T(x) = Ax for all x in Rn
A is mxn whose jth column is the vector T(ej) where ej is the jth column of InA = [T(e1) … T(en)]
Examine 1 point in each direction!!
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Existence Question for Transformations – “Onto"
Definition: A mapping T: Rn Rm is onto Rm if each b in Rm is the image of at least 1 x in Rn
i.e., range is all of the co-domain Equivalent: does Ax = b have a solution for all
b Theorem 1-4 Pivot in every row of A Theorem 1-12: T is onto iff columns of A span Rm
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Uniqueness Question for Transformations – “one-to-one”
Definition: A mapping T: Rn Rm is one-to-one if each b in Rm is the image of at most 1 x in Rn
Equivalent to Ax = b has one or no solution Pivot every column of A. Theorem 1-11: T is one-to-one iff T(x) = 0 has
only the trivial solution Theorem 1-12: T is one-to-one iff columns of A are
linearly independent
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Onto/One-to-One Example
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Example
T(x1, x2) = (3x1+x2, 5x1+7x2, x1+3x2)
a)Find standard matrix A of T.
b)Is T one-to-one?
c)Does T map R2 onto R3?
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