L: v --> w is a linear transformation. prove each of the following (a) ker l is a subspace of v. (b) range l is a subspace of w.

a) Assume that , and is a scalar (a real or complex number).

First. Let us prove that is not empty. This is easy because , by linearity. Here, stands for the zero vector of V, and stands for the zero vector of W.

Second. Let us prove that . By linearity

.

Then, .

Third. Let us prove that . Again, by linearity

.

And the statement readily follows.

b) Assume that and are in range of . Then, there exist such that and .

First. Let us prove that range of is not empty. This is easy because , by linearity.

Second. Let us prove that is on the range of .

.

Then, there exist an element such that . Thus is in the range of .

Third. Let us prove that is in the range of .

.

Then, there exist an element such that . Thus is in the range of .

Notice that in this second part of the problem we used the linearity in the reverse order, compared with the first part of the exercise.

Step-by-step explanation:

step-by-step explanation: so we look at the formula un = a + (n-1)d

un is the amount of seats

n is the nth term aka row number

a is the original sequence/row number

d is the difference between each term

so in this a=22 d=6 and n can remain as n because we are simply formjng a rule

lets put these in the equation

un=22+(n-1)(6)

this is your answer to part a

however

lets double check that this is correct, we'll try putting it into the second term

un=a+(n-1)d

u2=22+(2-1)(6)

u2=22+6

u2=28

as the question said the second row has 28 seats so now we know the formula we made is correct.

as for part b

un=a+(n-1)

its asking us to find a row with 100 seats.

remember un was the amount of seata so all we do is substitute this in.

un=100 a=22 d=6

un=a+(n-1)d

100=22+(n-1)(6)

so now we simplify to find n which is the row number.

100-22=22+(n-1)(6)-22

78=(n-1)(6)

78/6=(n-1)(6)/6

13=n-1

13+1=n-1+1

14=n

so now we know that in the 14th row there is 100 seats.

hope this and that you understood! its my first time using the app or anyone out on here : p its also me to revise so a bunch! ask more questions if you dont understand ☺