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| Date post: | 02-Jan-2016 |
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A man drops a ball from the top of a building.
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After ½ second, the ball has fallen 4 feet.
1/2 4
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After 1 second, the ball has fallen 16 feet.
1 16
1/2 4
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After 3/2 second, the ball has fallen 36 feet.
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1/2 4
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3/2
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After 2 seconds, the ball has fallen 64 feet.
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2 64
1/2 4
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3/2
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1 16
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Motion is described as a set of ordered pairs.
{ ( 1/2 , 4 ), ( 1 ,16 ), ( 3/2 , 36 ), ( 2 , 64 ) }
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1 16
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2 64
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Motion is described as a set of ordered pairs.
{ ( 1/2 , 4 ), ( 1 ,16 ), ( 3/2 , 36 ), ( 2 , 64 ) }
Sometimes there is a pattern, and we can write an equation:
d = 16 t2
t is time in seconds d is distance in feet
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1 16
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More generally, a function is defined as a set of ordered pairs.
{ ( 1/2 , 4 ), ( 1 ,16 ), ( 3/2 , 36 ), ( 2 , 64 ) }
When we write an equation for a function, the solutions (ordered pairs) define the function.
d = 16 t2
t is time in seconds d is distance in feet
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1 16
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2 64
1/2 4
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A function is defined as a set of ordered pairs.
{ ( 1/2 , 4 ), ( 1 ,16 ), ( 3/2 , 36 ), ( 2 , 64 ) }
The DOMAIN of the function = { 1/2 , 1 , 3/2 , 2 }
The RANGE of the function = { 4 ,16 , 36 , 64 }
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1 16
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2 64
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The DOMAIN of the function =1/2 1 3/2 2
The RANGE of the function = 4 16 36 64
The function is a mapping that relates everyDomain element t to a unique correspondingRange element, denoted f(t) and called the image of t
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1 16
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The DOMAIN of the function =1/2 1 3/2 2
The RANGE of the function = 4 16 36 64
The function is a mapping that relates everyDomain element t to a unique correspondingRange element, denoted f(t) and called the image of t
4 is the image of ½16 is the image of 136 is the image of 3/264 is the image of 2
4 = f ( ½ )16 = f ( 1 )36 = f ( 3/2 )64 = f ( 2 )
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1 16
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2 64
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The DOMAIN of the function =1/2 1 3/2 2
The RANGE of the function = 4 16 36 64
The function (of high school algebra fame) relates a set of real numbers to another set of real numbers.
Next we will examine a mapping that links a set of vectors to another set of vectors. In doing so, we use much of the same terminology that we used in the study of functions. A function is a type of mapping.
A farmer plans to purchase a herd of cows.
He considers 2 breeds:Purple cows and Brown cows
Each day a purple cow will eat 1 bale of hay and will produce 2 bottles of milk
Each day a brown cow will eat 2 bales of hay and will produce 3 bottles of milk
purple brown
A herd comprised of 50 purple and 70 brown cows will consume 190 bales of hay and produce 310 bottles of milk.
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purple brown
A herd comprised of 100 purple and 30 brown cows will consume 160 bales of hay and produce 290 bottles of milk.
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purple brown
A herd comprised of 80 purple and 150 brown cows will consume 380 bales of hay and produce 610 bottles of milk.
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purple brown
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The DOMAIN of the mapping:These vectors describe the composition of the herd, and this determines
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purple brown
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The DOMAIN of the mapping:These vectors describe the composition of the herd, and this determines The RANGE of the mapping:
These vectors describe the daily food intake and milk yield.
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purple brown
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Just as it is sometimes possible to find an equation to define a function, it is sometimes possible to produce a matrix to define a vector space mapping.
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purple brown
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Just as it is sometimes possible to find an equation to define a function, it is sometimes possible to produce a matrix to define a vector space mapping.
# bales of hay = 1 (# purple cows) + 2 (# brown cows)
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purple brown
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Just as it is sometimes possible to find an equation to define a function, it is sometimes possible to produce a matrix to define a vector space mapping.
# bales of hay = 1 (# purple cows) + 2 (# brown cows)
# bottles of milk = 2 (# purple cows) + 3 (# brown cows)
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purple brown
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Just as it is sometimes possible to find an equation to define a function, it is sometimes possible to produce a matrix to define a vector space mapping.
# bales of hay = 1 (# purple cows) + 2 (# brown cows)
# bottles of milk = 2 (# purple cows) + 3 (# brown cows)
brown
purple
bottles
bales
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#
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#
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purple brown
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brown
purple
bottles
bales
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#
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#
#
bottles
bales eg:
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purple brown
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brown
purple
bottles
bales
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#
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160 eg:
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,30
100 torelated vector thefind to
bymultiply
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purple brown
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brown
purple
bottles
bales
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#
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#
#
For every domain element ( a vector in R2 whose entries are the numbers of each breed of cow) there is a unique corresponding range element ( a vector in R2 whose entries are the numbers of bales consumed and bottles produced.)
A
vA
v
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purple brown
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brown
purple
bottles
bales
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#
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#
#A
30
100 of image the
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A
eg:
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