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kow [346]
2 years ago
12

Let T W Rn ! Rm be a linear transformation. Show that if T maps two linearly independent vectors onto a linearly dependent set,

then the equation T .x/ D 0 has a nontrivial solution. [Hint: Suppose u and v in Rn are linearly inde- pendent and yet T .u/ and T .v/ are linearly dependent. Then 495 c1T .u/ C c2T .v/ D 0 for some weights c1 and c2, not both zero. Use this equation.]
Mathematics
1 answer:
Mandarinka [93]2 years ago
6 0

Answer:

The solution proves that the equation has a non-trivial solution.  

Step-by-step explanation:

We want to show that the equation has nontrivial solutions for which c_{1} T(v_{1} ) + c_{2} T(v_{2} ) + c_{3} T(v_{3}) = 0

Let c_{i} be a set of non-zero numbers such that that c_{1} v_{1}  + c_{2} v_{2} + c_{3} v_{3}  = 0

Because T is a linear solution:

c_{1}T(v_{1} ) + c_{2} T (v_{2} ) + c_{3} T (v_{3} ) = T (c_{1} v_{1} ) + T(c_{2}v_{2} ) + T(c_{3} v_{3} )

                                            = T(c_{1}v_{1}  + c_{2} v_{2}  + c_{3} v_{3} )

                                            = T(0)

                                            = T (00)

                                            = 0T(0)

                                            = 0

This shows that T( v_{1} ) are linearly independent with the mapping resulting in a non-trivial solution.

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The graph represents the speed at which Pete’s marble travels.
AfilCa [17]

Answer:

B

Step-by-step explanation:

Here, we simply want to know who will reach the 6-foot mark sooner

Mathematically, we need to get the slope from the graph

Looking at the graph, we can see that the 3 ft mark was reached at exactly 2 seconds

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5 0
2 years ago
A yellow jacket can fly 4.5 meters in 9 seconds.find the unit rate in meters per second
snow_lady [41]
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6 0
2 years ago
Read 2 more answers
Simplify square root of (1-sin theta)(1+sin theta)
Vinil7 [7]
Recall the important identities:

i) (a-b)(a+b)= a^{2}- b^{2}, the difference of squares formula.

ii) sin^{2} x+cos^{2}x=1, for any angle x.

Then,

from i)

\sqrt{(1-sin(theta))(1+sin(theta))}= \sqrt{1- sin^{2}(theta) }

then, from ii), we have 

\sqrt{1- sin^{2}(theta) }= \sqrt{cos^{2}(theta) }=cos(theta)


Answer: cos(theta) 

6 0
2 years ago
Which is more​ economical: purchasing the economy size of a detergent at 7 kilograms for ​$7.15 or purchasing the regular size a
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8 0
2 years ago
Match the values associated with this data set to their correct descriptions
nordsb [41]
Put the numbers in order
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Q1 = (7 + 15) / 2 = 22/2 = 11 <== first quartile
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5 0
2 years ago
Read 2 more answers
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