Reaching everywhere with nothing spare

Mathematics · Linear Algebra · ages 18-19

Name ______________________ Date ____________

Mia says (1, 1) and (minus 1, minus 1) are independent because neither is the zero vector. Is Mia right?
Circle one: True False
What is the smallest number of vectors that can span all of R^3?
Answer: ______________
The equation c1*(1, 1) + c2*(2, 2) = (0, 0) holds with c2 = 4. What is c1?
Answer: ______________
Which of these pairs of vectors is independent in the plane?
- (1, 2) and (2, 4)
- (1, 0) and (0, 1)
- (3, 6) and (1, 2)
Why can the vector (5, 7) be written in only one way using (1, 0) and (0, 1)?
- Because (1, 0) and (0, 1) are linearly independent
- Because (5, 7) is longer than each of them
- Because coordinates must be whole numbers
- Because (1, 0) and (0, 1) are perpendicular to (5, 7)
What is the largest possible size of an independent set in the plane?
- 2
- 3
- 4
What does the span of the single nonzero vector (1, 2) look like in the plane?
- A line through the origin
- A circle around the origin
- The whole plane
- A single point
Why can the vector (5, 7) be written in only one way using (1, 0) and (0, 1)?
- Because (1, 0) and (0, 1) are independent
- Because coordinates must be whole numbers
- Because (5, 7) is longer than each of them
Write (7, 3) as a*(1, 1) + b*(1, -1). What is a?
Answer: ______________
Nadia says the set (1, 2), (2, 4), (0, 1) is a basis of the plane because it spans. What is wrong?
- It does not span the plane
- (2, 4) is twice (1, 2), so the set is dependent
- Three vectors can never span the plane

Answer key

For grown-ups. Fold this page away before handing over the rest.

False · Avoiding the zero vector is not enough: the second is minus 1 times the first.
3 · Fewer than 3 vectors span at most a plane, so 3 is the minimum, achieved by any basis.
-8 · Adding components gives c1 + 2*c2 = 0, so c1 = -8 when c2 = 4.
(1, 0) and (0, 1) · Only the standard basis pair is independent; each other pair holds a multiple.
Because (1, 0) and (0, 1) are linearly independent · If two combinations agreed, their difference would be a nontrivial dependence, which independence forbids.
2 · Two independent vectors already span the plane, so a third one always depends on them.
A line through the origin · All scalar multiples t*(1, 2) trace out the line through the origin in that direction.
Because (1, 0) and (0, 1) are independent · Two writings would differ by a dependence, which independence forbids.
5 · Adding components: a + b = 7 and a - b = 3, so a = 5 and b = 2.
(2, 4) is twice (1, 2), so the set is dependent · Two vectors already span the plane, so the third depends on them, and a basis needs independence.