Is W is a subspace of V? If not, state why. Assume that V has the standard operations. (Select all that apply.)
W = {(x1, x2, x3, 0): x1, x2, and x3 are real numbers}
V = R4

A) W is a subspace of V.

B) W is not a subspace of V because it is not closed under addition.

C) W is not a subspace of V because it is not closed under scalar multiplication.

A) W is a sub space of V

Step-by-step explanation:

given that and

W is a subspace of V because of the follwing.

let ∈ and ∈

now ∈

since are all elements of R

W is closed under vector adiition and scalar multiplication.

Hence W is a sub space of V.

use the power rule  

( a b ) n = a n b n

to distribute the exponent.

apply the product rule to  

3 x 2 y 3 . ( 3 x 2 ) 2 ( y 3 ) 2

apply the product rule to  

3 x 2 . 3 2 ( x 2 ) 2 ( y 3 ) 2

raise  

3

to the power of  

2 . 9 ( x 2 ) 2 ( y 3 ) 2 multiply the exponents in   ( x 2 ) 2

.

apply the power rule and multiply exponents,  

( a m ) n = a m n . 9 x 2 ⋅ 2 ( y 3 ) 2

multiply  

2   by   2 . 9 x 4 ( y 3 ) 2

multiply the exponents in  

( y 3 ) 2 .

apply the power rule and multiply exponents,  

( a m ) n = a m n . 9 x 4 y 3

⋅

2

multiply  

3   by   2 . 9 x 4 y 6

step-by-step explanation:

  y= -3x is the required function.

step-by-step explanation:

we have been given a table:

x             y

0             0

1             -3

3             -9

-2             6

we can see that the values of y is -3 times the value of x

hence, the function would be: y= -3x.

put the values of x will give the same y as in the table that means function is satisfying the given table.