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:

a; x = 0.5

step-by-step explanation:

-10x = -5

x = 0.5

u bout ugly as   a mf

step-by-step explanation: lol