A linear combination scales some vectors and adds the results: au + bv. Those two operations are the only ones linear algebra allows.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Span
The span of a set of vectors is every point you can reach by linear combinations of them. It is the set of all destinations.
Span of one vector
Scaling a single non-zero vector traces the whole line through it and the origin. That line is its span.
Span of two vectors
Two vectors pointing in different directions span the entire plane. Every point becomes reachable.
Unless they line up
If the second vector is a multiple of the first, it adds no new direction. The span stays a single line.
Every span holds the origin
Taking all coefficients zero always returns the zero vector. No span can avoid the origin.
A weighted sum
A linear combination scales each vector and adds the results. It is the only kind of expression the two allowed operations can build, which is why every construction in the subject is one.
Span is everything reachable
The span of a set is all the linear combinations of it. Two independent vectors in the plane span the whole plane; two parallel ones span only a line.
Spans are flat and contain the origin
A span is a point, a line, a plane or a higher flat through the origin. It always contains the zero vector, because scaling everything by zero is a permitted combination.
The question a system asks
Asking whether a system has a solution is asking whether a target vector lies in the span of the columns. That reformulation is what connects systems of equations to geometry.
Step 2: Try It Yourself
Tap and try it out.
- 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: 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 two vectors entry by entry.
Step 4: 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? 1 yes, 0 no.
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 5: 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.
- Two vectors span the plane unless one is a multiple of the other, which leaves only a line.