Question

Let v = (-18, 2, 3). Is v a member of the set span {(-3, 1, 0), (-7, 5, -2)}? Set up and solve a linear system to find the answer.

Answer 1

The vector v = (-18, 2, 3) is a member of the set spanned by {(-3, 1, 0), (-7, 5, -2)} as it can be expressed as a linear combination of the two provided vectors with coefficients a = 19/2 and b = -3/2.

Step-by-step explanation:

To determine if the vector v = (-18, 2, 3) is a member of the set spanned by {(-3, 1, 0), (-7, 5, -2)}, we need to see if v can be written as a linear combination of the two given vectors. This entails setting up a system of linear equations:

• -18 = -3a - 7b
• 2 = a + 5b
• 3 = 0a - 2b

The last equation instantly gives us b = -3/2. Substituting b into the second equation to find a:

2 = a + 5(-3/2)

2 = a - 15/2

a = 2 + 15/2

a = 4/2 + 15/2

a = 19/2

Now, substituting a back into the first equation to check consistency:

-18 = -3(19/2) - 7(-3/2)

-18 = -57/2 + 21/2

-18 = (-57 + 21) / 2

-18 = -36/2

-18 = -18

The system is consistent, and thus vector v is a member of the span of the two given vectors.