Question

Consider the following vectors:

→a =

5

−1

3

3

→b =

5

0

1

0

→c =

−10

3

−3

−7

For each of the following vectors, determine whether it is in span{→a, →b, →c}. If so, express it as a linear combination using a, b, and c as the names of the vectors above.

→v₁ =

5

−3

2

7

< Select an answer >

→v₂ =

2

7

6

−7

< Select an answer >

→v₃ =

30

−7

10

17

< Select an answer >

Answer 1

Let us consider a relation , where x,y,z are real numbers.

Then we have,

5x+5y-10z = 5................(i)

-x+3z = -3..................(ii)

3x+y-3z = 2.................(iii)

3x-7z = 7......................(iv)

Multiplying (ii) by 3 and adding with (iv) we get,

2z = -2

i.e., z = -1

Putting this in (ii) we get,

-x-3 = -3

i.e., -x = 3-3

i.e., -x = 0

i.e., x = 0

Putting this in (iii) we get,

0+y+3 = 2

i.e., y+3 = 2

i.e., y = 2-3

i.e., y = -1

Therefore, w₁ = 0*a+(-1)*y+(-1)*z.

Let us consider a relation , where x,y,z are real numbers.

Then we have,

5x+5y-10z = 2................(i)

-x+3z = 7..................(ii)

3x+y-3z = 6.................(iii)

3x-7z = -7......................(iv)

Multiplying (ii) by 3 and adding with (iv) we get,

2z = 14

i.e., z = 7

Putting this in (ii) we get,

-x+21 = 7

i.e., -x = 7-21

i.e., -x = -14

i.e., x = 14

Putting this in (iii) we get,

42+y-21 = 6

i.e., y+21 = 6

i.e., y = 6-21

i.e., y = -15

Therefore, w₂ = 14*a+(-15)*y+7*z.