A transformation T from ℝⁿ to ℝᵐ is linear when T(u + v) = T(u) + T(v) and T(cu) = cT(u) for all vectors and scalars. Linear transformations send the zero vector to zero and preserve every linear combination. Because each vector is a combination of the standard basis vectors, T is decided by where it sends e₁ through eₙ, and placing those images side by side gives its standard matrix. Pivots of that matrix then tell whether T is one-to-one and whether it is onto.
What you'll learn
Test whether a transformation is linear
Build the standard matrix of a linear transformation
Write rotations, reflections and projections as matrices
Decide whether a transformation is one-to-one or onto
A transformation T from ℝⁿ to ℝᵐ assigns an output vector T(x) in
ℝᵐ to every input x in ℝⁿ. Most such functions scramble vectors
arbitrarily. The ones that matter here respect the two operations of vector
algebra.
T is linear when, for all vectors u, v and every
scalar c,
T(u+v)=T(u)+T(v) and
T(cu)=cT(u).
Two consequences come at once. Taking c=0 gives T(0)=0:
a linear transformation fixes the origin. And applying both rules repeatedly,
Any x in ℝⁿ is a combination of the standard basis vectors:
x=x1e1+⋯+xnen. Linearity then gives
T(x)=x1T(e1)+⋯+xnT(en)
That right side combines the vectors T(ej) with the entries of
x as coefficients. That is exactly how
a matrix times a vector is defined.
A linear transformation is decided by where it sends e1 through
en, and those images, placed side by side, are its matrix:
A=[T(e1)T(e2)⋯T(en)]T(x)=Ax
This A is the standard matrix of T. Conversely, x↦Ax
is linear for any matrix, so linear transformations and matrices are two
descriptions of the same thing.
The unit square under T with T(e₁) = (2, 1) and T(e₂) = (−1, 1)
Two questions come up for every transformation, and the pivots of A answer
both.
T is one-to-one when different inputs always give different outputs.
That happens exactly when Ax=0 has only the trivial
solution: a pivot in every column.
T is onto ℝᵐ when every vector in ℝᵐ is an output. That happens exactly
when the columns of A span ℝᵐ: a pivot in every row.
For the T in Example 1, the matrix has two pivots in two rows but three
columns. It is onto ℝ² but not one-to-one; for instance
T(−2,1,0)=(0,0)=T(0).
T(e1)=(3,1) and T(e2)=(−1,4) become the columns.
A linear T has T(e1)=(1,4) and T(e2)=(2,−1). Find T(3,2).
Answer
(7,10)
Full solution
T(3,2)=3T(e1)+2T(e2)=(3,12)+(4,−2).
Find the matrix that reflects the plane across the x-axis, and apply it to (4,5).
Answer
[100−1]; (4,−5)
Full solution
e1 stays put and e2 flips to (0,−1).
Is T(x,y)=(x+y,2) linear?
Answer
No
Full solution
T(0,0)=(0,2), not the zero vector.
Find the standard matrix of T(x,y)=(x,y,x+y) from ℝ² to ℝ³. Is T one-to-one?
Answer
101011; yes
Full solution
The columns are T(e1)=(1,0,1) and T(e2)=(0,1,1). Both columns hold pivots, so the transformation is one-to-one.
Is the transformation in exercise 5 onto ℝ³?
Answer
No
Full solution
A 3×2 matrix has at most two pivots, so one of its three rows has none. Two vectors cannot span ℝ³.
What single matrix rotates the plane by 90° four times in a row?
Answer
The identity matrix
Full solution
Four quarter turns make a full turn, which moves nothing. As a check, R2=−I, so R4=(−I)2=I.
A linear T sends (1,1) to (3,0) and (1,−1) to (1,2). Find T(e1).
Hint
Write e1 as a combination of (1,1) and (1,−1).
Answer
(2,1)
Full solution
e1=21(1,1)+21(1,−1), so T(e1)=21(3,0)+21(1,2)=(2,1).
A linear transformation from ℝ⁵ to ℝ³ has a standard matrix with three pivots. Is it one-to-one? Onto?
Answer
Onto, but not one-to-one
Full solution
Three pivots fill all three rows, so the columns span ℝ³. Five columns with three pivots leave two free variables, so nonzero inputs reach 0.
For T(x,y)=(x+2y,3x), a student finds T(e1)=(1,3) and T(e2)=(2,0), writes them as rows, and gets [1230]. What went wrong?
Hint
Multiply the student’s matrix by e1.
Answer
The images belong in columns: [1320].
Full solution
The student’s matrix sends (1,0) to its first column, (1,2), but T(1,0)=(1,3). Rows give the transpose, a different transformation. With the images as columns, the matrix sends e1 to (1,3) and e2 to (2,0), as it must.
Frequently asked questions
What makes a transformation linear?
It must satisfy T(u + v) = T(u) + T(v) and T(cu) = cT(u) for every pair of vectors and every scalar. Together these say T preserves linear combinations.
What is the standard matrix of a linear transformation?
The matrix whose columns are T(e₁), …, T(eₙ). Multiplying by it does exactly what T does.
Is a translation linear?
No. A linear transformation sends 0 to 0, and a translation moves the origin.
When is a linear transformation one-to-one?
When its standard matrix has a pivot in every column, so the columns are independent and Ax = 0 has only the trivial solution.
When is a linear transformation onto?
When its standard matrix has a pivot in every row, so the columns span the whole target space ℝᵐ.