Let u1 and u2 be the vectors: a) Find a vector v such that {u1,u2,v} is linearly

Question

Let u1 and u2 be the vectors:

a) Find a vector v such that {u1,u2,v} is linearly independent.
b) Find a vector v such that {u1,u2,v} is linearlydependent.

c) Find a vector v such that {u1,u2,v} spans IR^3

d) Find a vector v such that {u1,u2,v} does not span IR^3

Let u1 and u2 be the vectors: u1 = [1 2 3] u2 = [2 1 0] Find a vector v such that {ul, u2, v} is linearly independent. Find a vector v such that {u1, u2, v} is linearly dependent. Find a vector v such that {u1, u2, v} spans R3. Find a vector v such that {u1, u2, v} does not span R3.

Explanation / Answer

u1 = <1,2,3>

u2 = <2,1,0>

a.

v would have three components, v = <v1, v2, v3>

if we put the u1, u2, and v into a matrix and found the determinant, the vectors would be independent if the determinant was NOT 0.

| 1 2 v1 |

| 2 1 v2 |

| 3 0 v3 |

= v3 + 6v2 + 0v1 - 3v1 - 4v3 - 0v2 not= 0

= v3 + 6v2 - 3v1 - 4v3 not= 0

= -3v3 + 6v2 - 3v1 not= 0  

An example of this would be v=<v1,v2,v3> = <1, 2, 1>

b.

v could be <2,4,6> or any constant multiple of one of the vectors.

c. The answer to (a) will span r3

d. The answer to (b) will not span r3

Related Questions

Navigate

• Browse (All)
• Browse L
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.