A linear combination scales some vectors and adds the results, au + bv, and those are the only two operations linear algebra allows. The span of a set is every point its combinations can reach. One non-zero vector spans a line; two in different directions span the whole plane; two that line up still span only a line. Every span contains the origin.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Span
The span is the set of all destinations. Ask: starting from the origin, walking only along these arrows and their multiples, where can I get? For ⟨2, 1⟩ and ⟨1, 3⟩ the answer is everywhere.
One vector
Scaling ⟨1, 2⟩ by every number traces the line through it and the origin: ⟨2, 4⟩, ⟨−1, −2⟩, ⟨0.5, 1⟩. That line is its span, and nothing off the line is reachable.
Two directions
Can ⟨7, 8⟩ be reached from ⟨2, 1⟩ and ⟨1, 3⟩? a⟨2, 1⟩ + b⟨1, 3⟩ = ⟨7, 8⟩ means 2a + b = 7 and a + 3b = 8. Solve: b = 7 − 2a, so a + 21 − 6a = 8, a = 2.6, b = 1.8.
Check the weights
2.6⟨2, 1⟩ + 1.8⟨1, 3⟩ = ⟨5.2 + 1.8, 2.6 + 5.4⟩ = ⟨7, 8⟩. Any target gives a solvable system, because the two directions differ. The span is the plane.
Unless they line up
If the second vector is a multiple of the first, it adds no new direction, and the system for a target off the line has no solution. Check for multiples first: compare the ratios of the entries.
Every span holds the origin
Taking every coefficient zero returns the zero vector, so no span can avoid the origin. A line that misses the origin is not the span of anything.
Step 2: Try It Yourself
Tap and try it out.
u = ⟨2, 1⟩, v = ⟨1, 3⟩. Decide whether they span the plane, then find the weights that reach ⟨7, 8⟩ and check them
- 11/2 ≠ 3/1: neither is a multiple of the othertwo different directions
Step 0 of 4
- i-hat lands on(2, 1)
- j-hat lands on(1, 3)
- Determinant5
The shaded parallelogram is the image of the unit square, and its area is 5. That is exactly what the determinant measures.
Step 3: In Real Life
Mixing paint
Red and blue paint span every purple. Add yellow and you span far more. What colors you can reach from your starting tubes is a span.
Step 4: Watch an Example
One step at a time.
Watch Rafael Test a Span
Rafael asks whether ⟨1, 2⟩ and ⟨3, 6⟩ span the whole plane.
- Step 1
He compares the vectors entry by entry and notices the second is exactly 3 times the first.
Step 5: Your Turn
Practice makes it stick.
The Combination
Problem 1 of 2
2⟨1, 3⟩ + 4⟨2, 0⟩. What is the first entry?
The Line
Problem 2 of 2
Two vectors that are multiples of each other. What is the dimension of their span?
What Can You Reach
1 of 8
3⟨2, 1⟩ + 2⟨0, 5⟩. What is the second entry?
2 of 8
The span of a single non-zero vector. What is its dimension?
3 of 8
⟨1, 0⟩ and ⟨0, 1⟩. What is the dimension of their span?
4 of 8
⟨2, 4⟩ and ⟨1, 2⟩. What is the dimension of their span?
5 of 8
Does the origin belong to every span?
6 of 8
The span of the zero vector alone. What is its dimension?
7 of 8
Sort each pair of vectors by what they span.
Tap something to move it.
- Empty
- Empty
8 of 8
0⟨5, 7⟩ + 0⟨2, 9⟩. What is the first entry?
Step 6: Quick Check
Show what you know.
Question 1 of 2
⟨4, 8⟩ and ⟨1, 2⟩. What is the dimension of their span?
Question 2 of 2
What is the span of a set of vectors?
What You Learned
- A linear combination scales vectors and adds them, and nothing else is allowed.
- The span is every point those combinations can reach, and it always holds the origin.
- Two vectors in different directions span the plane; a multiple adds no new direction and leaves a line.
- Reaching a target means solving one equation per entry for the weights.