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: Picture It
Start with something you already know.
A drawing toy with two knobs
- A drawing toy has two knobs and a pen that starts in the middle of the screen.
- One turn of the first knob, u, slides the pen 2 right and 1 up.
- One turn of the second knob, v, slides the pen 1 right and 3 up.
- Turning a knob backward slides the pen the opposite way, and part of a turn counts too.
- Now picture a broken toy whose two knobs both slide the pen along the same slanted track.
A linear combination is one setting of the knobs: so many turns of u, so many turns of v.
The span is every spot the pen can reach. Two knobs that slide different ways reach the whole screen; two that slide along one track reach only that track.
Where it breaks: a real screen has an edge, but the plane and the number of turns go on forever.
The first knob is u = ⟨2, 1⟩ and the second is v = ⟨1, 3⟩:
| Turns of u, turns of v | The pen lands at |
|---|---|
| 1 and 0 | ⟨2, 1⟩ |
| 1 and 1 | ⟨3, 4⟩ |
| 1 and −1 | ⟨1, −2⟩ |
| 2.6 and 1.8 | ⟨7, 8⟩ |
Every row is au + bv for some a and b. With these two knobs, some setting reaches any spot you name.
Step 2: Let's Learn
Read it, or press Listen and follow the words.
Try it first
A robot on a grid has two moves: a step of ⟨1, 0⟩, one square right, and a step of ⟨1, 1⟩, one square right and one up. It takes only these steps, forward, and ends at ⟨5, 3⟩. How many ⟨1, 0⟩ steps did it take?
steps
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
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 3: 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 4: 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 5: Watch an Example
One step at a time.
Watch Ana Reach a Target
u = ⟨1, 1⟩ and v = ⟨1, −1⟩. Ana asks whether these two span the plane, and then finds the weights a and b with au + bv = ⟨5, 1⟩.
- Step 1
Ana tests for a multiple first. Dividing v’s entries by u’s gives 1/1 = 1 and −1/1 = −1. The two ratios differ, so no single number turns u into v: they point in two different directions.
Watch Maya Find the Weights, Then Finish It Yourself
u = ⟨1, 2⟩ and v = ⟨3, 1⟩. Maya wants the weights a and b with au + bv = ⟨9, 8⟩. She starts; the rest is yours.
- Step 1
The ratio test comes first. Dividing v’s entries by u’s gives 3/1 = 3 and 1/2 = 0.5, which differ, so u and v point two different ways and their span is the plane. The target can be reached.
Watch Dev Meet Two Vectors That Line Up
u = ⟨2, −1⟩ and v = ⟨−6, 3⟩. Dev asks what they span, and whether ⟨4, −2⟩ and ⟨4, 1⟩ can be reached.
- Step 1
Dividing v’s entries by u’s gives −6/2 = −3 and 3/(−1) = −3. The ratios agree, so v = −3u: the same line as u, pointing the opposite way.
Step 6: Your Turn
Practice makes it stick.
Two Trails
Problem 1 of 15
A park’s trails run in only two directions. One leg of trail A takes a hiker 2 km east and 1 km north, ⟨2, 1⟩. One leg of trail B takes her 1 km east and 2 km north, ⟨1, 2⟩. She can walk any number of legs of each, in either direction, parts of legs included. How many legs of trail A does she walk to reach a lookout at ⟨7, 5⟩?
legs
Gift Boxes
Problem 2 of 15
A shop sells two gift boxes. Box A holds 2 mugs and 1 candle; box B holds 1 mug and 1 candle. Box A costs $12 and box B costs $9. An office wants exactly 10 mugs and 7 candles, in whole boxes. What do the boxes cost in all, in dollars?
dollars
Combine and Reach
Scale and add one entry at a time. Where weights are asked for, write one equation per entry.
Problem 3 of 15
u = ⟨2, −1⟩ and v = ⟨1, 3⟩. What is the second entry of 3u + 2v?
Problem 4 of 15
⟨−2, 6⟩ = c⟨1, −3⟩ for one number c. What is c?
Problem 5 of 15
a⟨1, 0⟩ + b⟨2, 1⟩ = ⟨9, 3⟩. What is a?
Problem 6 of 15
a⟨3, 1⟩ + b⟨1, 2⟩ = ⟨5, 5⟩. What is b?
Problem 7 of 15
What do ⟨4, 1⟩ and ⟨2, 3⟩ span?
Whose Step?
One line on each page is a slip. Find it, then put it right.
Problem 8 of 15
Leo decided what u = ⟨3, −6⟩ and v = ⟨1, −2⟩ span. One line is a slip. Which?
Leo’s span of ⟨3, −6⟩ and ⟨1, −2⟩
- 1u = ⟨3, −6⟩ and v = ⟨1, −2⟩
- 2there are two vectors
- 3so there are two directions
- Answerspan = the whole plane
Problem 9 of 15
Worked correctly, what do ⟨3, −6⟩ and ⟨1, −2⟩ span?
Problem 10 of 15
Sam wanted a and b with a⟨1, 2⟩ + b⟨1, −1⟩ = ⟨5, 4⟩. One line is a slip. Which?
Sam’s weights for ⟨5, 4⟩ from ⟨1, 2⟩ and ⟨1, −1⟩
- 1a + b = 5 and 2a − b = 4
- 2b = 5 − a
- 32a − 5 − a = 4
- 4a = 9
- Answerb = −4
Problem 11 of 15
Worked correctly, what is a in a⟨1, 2⟩ + b⟨1, −1⟩ = ⟨5, 4⟩?
Pick the Move
Choose the move that gets the answer. You do not need to work either question all the way through.
Problem 12 of 15
You must decide whether ⟨1, 4⟩ and ⟨2, 7⟩ span the whole plane. Which move settles it?
Problem 13 of 15
To reach ⟨4, 9⟩ from ⟨1, 2⟩ and ⟨1, 3⟩, you have written a + b = 4 and 2a + 3b = 9. Which move comes next?
Same Span?
Judge each one without solving a system. Select every entry that fits.
Problem 14 of 15
Select every set whose span is the same line as the span of ⟨1, −2⟩ alone.
Problem 15 of 15
Select every pair whose span is the whole plane.
Step 7: 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?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
For which number k do ⟨2, k⟩ and ⟨3, 6⟩ span only a line?
Stretch 2 of 2
Every span holds the origin, because all-zero weights reach it. For u = ⟨1, 2⟩ and v = ⟨2, 4⟩ there is another way: au + bv = ⟨0, 0⟩ with a = 2. What is b?
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.
For the grown-up
The misconception this lesson targets is that the span is decided by counting vectors: two vectors, so the whole plane. A student who holds it never tests for a multiple, so ⟨3, −6⟩ and ⟨1, −2⟩ look like two directions when they are one line. The cheap habit that settles it is the ratio test, dividing matching entries before anything else, and saying "directions, not vectors" aloud. A second, quieter slip is reading the target’s entries as the weights, which is true only for ⟨1, 0⟩ and ⟨0, 1⟩; one equation per entry, then a check that the weights rebuild the target, catches it every time. The knob picture helps because a broken toy whose knobs slide the same way is easy to imagine, and it is exactly two vectors on one line.