A matrix does not merely store numbers. It moves every point of the plane, and the movement is completely determined by where two vectors go.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Two vectors decide everything
Once you know where ⟨1, 0⟩ and ⟨0, 1⟩ land, every other vector follows, because every vector is a combination of those two.
The columns
The first column of the matrix is where ⟨1, 0⟩ lands. The second column is where ⟨0, 1⟩ lands. That is what a matrix is.
The grid moves too
A linear transformation keeps grid lines straight, parallel and evenly spaced. It may tilt and stretch them but never bends them.
The origin stays put
A times the zero vector is always zero. Linear transformations never slide the plane sideways.
Building one
To construct a transformation, decide where the two basis vectors should go and write those as the columns.
The columns are where the basis vectors land
The first column of A is where the vector ⟨1, 0⟩ goes, the second where ⟨0, 1⟩ goes. Knowing that, you can write down the matrix of any transformation you can describe geometrically.
Basis images determine everything
Because every vector is a combination of the basis vectors, and the map is linear, the images of the basis fix the whole transformation. That is why a finite matrix describes an infinite map.
Lines stay lines
A linear transformation sends lines to lines and keeps the origin fixed. Grids of parallel lines remain parallel and evenly spaced, which is what the word linear is recording.
Translations are not linear
Shifting the plane moves the origin, so it is not a linear transformation. Graphics systems handle it with an extra coordinate, which is why they use 4×4 matrices for three-dimensional work.
Step 2: Try It Yourself
Tap and try it out.
- i-hat lands on(1, 0)
- j-hat lands on(1, 1)
- Determinant1
- v lands on(3, 1)
The shaded parallelogram is the image of the unit square, and its area is 1. That is exactly what the determinant measures.
Step 3: Watch an Example
One step at a time.
Watch Fatima Build a Transformation
Fatima wants the transformation sending ⟨1, 0⟩ to ⟨0, 1⟩ and ⟨0, 1⟩ to ⟨−1, 0⟩.
- Step 1
She writes the image of the first basis vector as the first column: ⟨0, 1⟩.
Step 4: Your Turn
Practice makes it stick.
The Landing
Problem 1 of 2
A matrix has first column ⟨4, 3⟩. Where does ⟨1, 0⟩ land? Enter the second entry.
The Origin
Problem 2 of 2
Where does the origin land under any linear transformation? Enter the first entry.
Move the Plane
1 of 8
The first column of a matrix is ⟨2, 5⟩. Where does ⟨1, 0⟩ land? Enter the first entry.
2 of 8
The second column is ⟨0, 7⟩. Where does ⟨0, 1⟩ land? Enter the second entry.
3 of 8
Does a linear transformation ever bend a straight line? 1 yes, 0 no.
4 of 8
The identity matrix. Where does ⟨3, 4⟩ land? Enter the first entry.
5 of 8
A matrix with both columns ⟨1, 1⟩. Does it flatten the plane to a line? 1 yes, 0 no.
6 of 8
How many vectors decide a transformation of the plane completely?
7 of 8
Sort each property by whether a linear transformation preserves it.
Tap something to move it.
- Empty
- Empty
8 of 8
A matrix with columns ⟨2, 0⟩ and ⟨0, 2⟩. By what factor does it stretch every vector?
Step 5: Quick Check
Show what you know.
Question 1 of 2
The first column of a matrix is ⟨0, 6⟩. Where does ⟨1, 0⟩ land? Enter the second entry.
Question 2 of 2
What do the columns of a matrix tell you?
What You Learned
- A matrix is a transformation that moves every point of the plane.
- Its columns are exactly where the two basis vectors land, and everything else follows.
- Grid lines stay straight, parallel and evenly spaced, and the origin never moves.