CLASS NOTES FOR LINEAR MATH

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CLASS NOTES FOR LINEAR MATH

• Section 3.3
• LINEAR (IN)DEPENDENCE

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RECALL
A set of matrices S A1, A2, , Ak is linearly
dependent if there are constants c1, c2, , ck
that are NOT ALL zero and c1A1 c2A2 ckAk
0. Otherwise S is called linearly
independent. This means that if S is linearly
independent and c1A1 c2A2 ckAk 0 then
ALL of the cs are zero.
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EXAMPLE 1
Determine if x1, x2-4x, and 2x23x-5 are
linearly independent in P2.
a(x1) b(x2-4x)
c(2x23x-5) 0 a 0b - 5c 0 a - 4b 3c
0 0a b 2c 0
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EXAMPLE 1
Determine if x1, x2-4x, and 2x23x-5 are
linearly independent in P2.
a(x1) b(x2-4x)
c(2x23x-5) 0 a 0b - 5c 0 a - 4b 3c
0 0a b 2c 0
Notice that row operations do not affect a column
of zeros and that the end result has a non-zero
determinant for the coefficient matrix.
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EXAMPLE 1
Determine if x1, x2-4x, and 2x23x-5 are
linearly independent in P2. (A simpler method.)
a(x1)
b(x2-4x) c(2x23x-5) 0 a 0b - 5c 0 a - 4b
3c 0 0a b 2c 0
ANSWER YES
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EXAMPLE 2
Determine if f(x) 1, g(x) sin2(x) and h(x)
cos(2x) are linearly independent (in C1)
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EXAMPLE 2
Determine if f(x) 1, g(x) sin2(x) and h(x)
cos(2x) are linearly independent (in C1)
Since a(1) bsin2(x) ccos(2x) 0 is an
equation of functions, it needs to hold for every
value in the domain of those functions, i.e. for
every real number. We only need three values to
determine a, b, and c, but we also need to pick
them wisely. x 0, 2? and 4? for example would
give little information.
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EXAMPLE 2
Determine if f(x) 1, g(x) sin2 x and h(x)
cos(2x) are linearly independent (in C1)
a(1) bsin2 x ccos(2x) 0
Let x 0 gt a c 0 Let x p/4 gt 2a b
0 Let x p/2 gt a b - c 0
c c, a -c and b 2c
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EXAMPLE 2
Determine if f(x) 1, g(x) sin2 x and h(x)
cos(2x) are linearly independent (in C1)
-c(1) 2csin2 x ccos(2x) 0
For c 1, then -1 2sin2 x cos(2x) 0 Or
cos(2x) 1 2sin2 x This is one of the double
angle identities from trigonometry, so we know
that this equation holds in general. However, if
we didnt remember this identity, how would we
know that other choices of x wouldnt show that
the constants a, b and c would all have to be
zero?
DEPENDENT THIS IS NOT A RELIABLE TEST FOR
DEPENDENCE, JUST INDEPENDENCE
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EXAMPLE 2
Determine if f(x) 1, g(x) sin2 x and h(x)
cos(2x) are linearly independent (in C1)
-c(1) 2csin2 x ccos(2x) 0
cos(2x) 1 2sin2 x By the way this also
shows that Spanf, g Spanf, g, h
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EXAMPLE 3
Determine if f(x) sin x, g(x) sin(2x) and h(x)
cos(2x) are linearly independent (in C1)
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EXAMPLE 3
Determine if f(x) sin x, g(x) sin(2x) and h(x)
cos(2x) are linearly independent (in C1)
asin x bsin(2x) ccos(2x) 0
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EXAMPLE 3
Determine if f(x) sin x, g(x) sin(2x) and h(x)
cos(2x) are linearly independent (in C1)
asin x bsin(2x) ccos(2x) 0
Let x 0 ccos(2x) 0 so c 0 Let x ??/2
a c 0 so a 0 0sinx bsin(2x) 0cos(2x)
0 gives b 0.
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EXAMPLE 3
Determine if f(x) sin x, g(x) sin(2x) and h(x)
cos(2x) are linearly independent (in C1)
asin x bsin(2x) ccos(2x) 0
Let x 0 ccos(2x) 0 so c 0 Let x ??/2
a c 0 so a 0 0sinx bsin(2x) 0cos(2x)
0 gives b 0. They are linearly independent.
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DEFINITION OF A BASIS

• A set B in a vector space V is called a basis for
  V if
• Span B V
• B is linearly independent

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EXAMPLES OF BASES

• A (the standard) basis for P2 is B 1, x, x2
• The standard basis for R2 is B (1, 0), (0,
  1)
• A different basis for R2 is B (-1,2),
  (5,8)
• Is (1,2,3), (4,5,6), (7,8,9) a basis for R3?

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End of Section 3.3