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addi92
11.02.2020 •
Mathematics
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.]
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Ответ:
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](/tpl/images/0507/2717/2cca3.png)
Let
be a set of non-zero numbers such that that ![c_{1} v_{1} + c_{2} v_{2} + c_{3} v_{3} = 0](/tpl/images/0507/2717/a570d.png)
Because T is a linear solution:
=![T(c_{1}v_{1} + c_{2} v_{2} + c_{3} v_{3} )](/tpl/images/0507/2717/9882e.png)
=![T(0)](/tpl/images/0507/2717/c0db4.png)
=![T (00)](/tpl/images/0507/2717/f021c.png)
=![0T(0)](/tpl/images/0507/2717/7612d.png)
= 0
This shows that
are linearly independent with the mapping resulting in a non-trivial solution.
Ответ:
There are 20 children
Step-by-step explanation:
The number who drink water with their lunch is equal to the number of children times the percentage who drink water with their lunch.
Drink water = children * 15%
3 = children * .15
Divide each side by .15
3/.15 = children * .15/.15
20 = children