Cogito
Linear Algebra · Chapter 1 · Lesson 2
Linear Combinations and Span
Everything you can reach from where you started.
12 problems · about 23 minutes · N-VM.B.4, N-VM.B.5
What this lesson teaches
The student forms linear combinations and describes the span of a set of vectors.
- A linear combination scales vectors and adds them, and nothing else is allowed.
- The span is every point those combinations can reach.
- Two vectors span the plane unless one is a multiple of the other, which leaves only a line.
Warm Up
Straightforward practice. Get the method working first.
5 problems⟨4, 8⟩ and ⟨1, 2⟩. What is the dimension of their span?
Answer 1
Why 1.
What is the span of a set of vectors?
Answer Every point reachable by scaling and adding those vectors.
Why Everything you can reach.
3⟨2, 1⟩ + 2⟨0, 5⟩. What is the second entry?
Answer 13
Why 3 + 10.
The span of a single non-zero vector. What is its dimension?
Answer 1
Why A line.
⟨1, 0⟩ and ⟨0, 1⟩. What is the dimension of their span?
Answer 2
Why They reach the whole plane.
Build It Up
The same ideas with more to keep track of.
3 problems⟨2, 4⟩ and ⟨1, 2⟩. What is the dimension of their span?
Answer 1
Why One is a multiple of the other.
Does the origin belong to every span? 1 yes, 0 no.
Answer 1
Why Take all coefficients zero.
The span of the zero vector alone. What is its dimension?
Answer 0
Why Scaling zero gives only zero.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each pair of vectors by what they span.
Answer The whole plane: ⟨1, 0⟩ and ⟨0, 1⟩, ⟨3, 1⟩ and ⟨1, 4⟩ · Only a line: ⟨1, 1⟩ and ⟨2, 2⟩, ⟨2, 6⟩ and ⟨1, 3⟩
Why Check whether one is a multiple of the other.
0⟨5, 7⟩ + 0⟨2, 9⟩. What is the first entry?
Answer 0
Why Everything scales to nothing.
The Combination: 2⟨1, 3⟩ + 4⟨2, 0⟩. What is the first entry?
Answer 10
Why 10.
The Line: Two vectors that are multiples of each other. What is the dimension of their span?
Answer 1
Why 1.