Linear Transformations
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Linear Transformations
A transformation (or function or mapping) T from ℝn to ℝm is a rule
that assigns to each vector x in ℝn a vector T (x) in ℝm .
The set ℝn is called the domain of T, and ℝm is called the codomain of T.
The notation T: ℝn → ℝm says the domain of T is ℝn and codomain is ℝm .
For x in ℝn , the vector T (x) in ℝm is called the image of x.
The set of all images T (x) is called the range of T.
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Linear Transformations
For each x in ℝn , T(x) is computed as Ax, where A is an mxn matrix.
For simplicity, we denote such a matrix transformation by x↦Ax.
The domain of T is ℝn when A has n columns and the codomain of T is ℝm
when each column of A has m entries.
So an mxn matrix transforms vectors from ℝn into vectors from ℝm.
Here are a few examples of transformation matrices:
1 2
A1  

0 1
1  1
A2  

1 1 
1 2
A3  3 4
5 6
1 2 1 4 2


A 4  0 0 1 3 5
1 2 3 1 0
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1 2
A1  

0 1
This matrix is 2x2, so it transforms vectors
from ℝ2 into (other) vectors from ℝ2.
To see what this matrix does, we can where it takes a few specific vectors.
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1 2
A1  

0 1
This matrix is 2x2, so it transforms vectors
from ℝ2 into (other) vectors from ℝ2.
To see what this matrix does, we can where it takes a few specific vectors.
1 2 1 1
0 1  0  0

    
1 2 0 2

    
0 1 1 1
1 2 1 3
0 1  1  1

    
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1 2
A1  

0 1
This matrix is 2x2, so it transforms vectors
from ℝ2 into (other) vectors from ℝ2.
To see what this matrix does, we can where it takes a few specific vectors.
1 2 1 1
0 1  0  0

    
1 2 0 2

    
0 1 1 1
1 2 1 3
0 1  1  1

    


x  A1x
Domain (ℝ2)
Range (ℝ2)
This is a shear transformation.
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1  1
A2  

1 1 
This matrix is 2x2, so it transforms vectors
from ℝ2 into (other) vectors from ℝ2.
To see what this matrix does, we can where it takes a few specific vectors.
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1  1
A2  

1 1 
This matrix is 2x2, so it transforms vectors
from ℝ2 into (other) vectors from ℝ2.
To see what this matrix does, we can where it takes a few specific vectors.
1  1 1 1
1 1   0  1

    
1  1 0  1

    
1 1  1  1 
1  1 1 0
1 1   1  2

    
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1  1
A2  

1 1 
This matrix is 2x2, so it transforms vectors
from ℝ2 into (other) vectors from ℝ2.
To see what this matrix does, we can where it takes a few specific vectors.
1  1 1 1
1 1   0  1

    
1  1 0  1

    
1 1  1  1 
1  1 1 0
1 1   1  2

    


x  A2x
Domain (ℝ2)
Range (ℝ2)
This matrix is a combination of a rotation through 45° and a stretch by a factor of √2.
We will have more to say about this type of matrix when we cover Chapter 5.
Check out your textbook for more discussion of 2x2 transformation matrices.
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1 2
A3  3 4
5 6
This matrix is 3x2, so it transforms vectors
from ℝ2 into vectors from ℝ3.
This transformation takes a vector from ℝ2 and maps it to a vector in ℝ3. There is more we
can say though. The range of this transformation is not the entire 3-dimensional ℝ3 space.
The images must be in a subset of ℝ3 that has dimension (at most) 2 – a plane.
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Linear Transformations
1 2
A3  3 4
5 6
This matrix is 3x2, so it transforms vectors
from ℝ2 into vectors from ℝ3.
This transformation takes a vector from ℝ2 and maps it to a vector in ℝ3. There is more we
can say though. The range of this transformation is not the entire 3-dimensional ℝ3 space.
The images must be in a subset of ℝ3 that has dimension (at most) 2 – a plane.


x  A3x
Domain (ℝ2)
Range (plane in ℝ3)
The images all lie on a plane. The Range can’t have a larger dimension than the Domain.
We will see this as a more general rule later, but for now we need to know the concepts of
ONE-TO-ONE and ONTO.
This transformation is not ONTO because it does not span all of ℝ3.
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DEFINITIONS:
A mapping T:ℝn↦ℝm is said to be ONTO (or surjective) if each b in
ℝm is the image of at least one x in ℝn.
Domain
ℝn
T
Range is
All of
T is onto
ℝm
T
Domain
ℝn
Range is a
subspace of
ℝm
T is not onto
A mapping T:ℝn↦ℝm is said to be ONE-TO-ONE (or injective) if each
b in ℝm is the image of at most one x in ℝn.
T
T is one-to-one
T
T is not one-to-one
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A couple of quick tests to see if a transformation is one-to-one or onto:
More Columns than Rows – it can’t be One-to-One
More Rows than Columns – it can’t be Onto
More precisely:
A transformation is onto iff the columns of A span ℝm.
A transformation is one-to-one iff the columns are linearly independent.
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1 2 1 4 2


A 4  0 0 1 3 5
1 2 3 1 0
This matrix is 3x5, so it transforms vectors
from ℝ5 into (other) vectors from ℝ3.
To see what this matrix does, we can where it takes a few specific vectors.
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1 2 1 4 2


A 4  0 0 1 3 5
1 2 3 1 0
This matrix is 3x5, so it transforms vectors
from ℝ5 into (other) vectors from ℝ3.
To see what this matrix does, we can where it takes a few specific vectors.
1
1 2 1 4 2 0 1
0 0 1 3 5     0
 0  

1 2 3 1 0 0 1
0
0
1 2 1 4 2 1 2
0 0 1 3 5     0
 0  

1 2 3 1 0 0 2
0
0
1 2 1 4 2 0 1
0 0 1 3 5     1
 1  

1 2 3 1 0 0 3
0
This one is a bit harder to visualize, but we are starting with
vectors from ℝ5, and mapping them to vectors in ℝ3.
The transformation is definitely not ONE-TO-ONE because the
dimension of the range (at most 3) is certainly lower than the
domain (5).
The transformation will be ONTO as long as the set of column
vectors in the matrix spans all of ℝ3. This can be checked in the
usual way by row reducing the matrix and seeing that there are
3 pivot positions in the RREF form (see below).
RREF for this matrix is 1 2 0 0  19 
3

(check this yourself!) 0 0 1 0
3 
0 0 0 1

4 
3 
 etc
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Linear Transformations
So far we have seen a few linear transformations, but what makes them LINEAR?
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Linear Transformations
So far we have seen a few linear transformations, but what makes them LINEAR?
To be linear, a transformation must have the following properties:
 


T(u  v)  T(u)  T(v) For any vectors u and v in the domain of T


For all scalars c and every vector u in the domain of T
T(cu)  cT(u)
The basic idea is that for vector addition and scalar multiplication, the results are the
same if you perform the operation before or after you apply the transformation.
An important special case of the scalar multiplication rule is that


T(0)  0
This gives an easy way to test a transformation for linearity.
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Are the following transformations linear?
𝑎) 𝑇 𝑥, 𝑦 = (2𝑥, 𝑥 + 𝑦)
𝑏) 𝑇 𝑥, 𝑦 = (𝑥 − 3𝑦, 𝑥𝑦)
𝑐) 𝑇 𝑥, 𝑦, 𝑧 = (𝑥, 𝑦, 0)
𝑑) 𝑇 𝑥, 𝑦, 𝑧 = (2𝑥, 2𝑦, 2)
e) 𝑇 𝑥, 𝑦, 𝑧, 𝑤 = (2𝑥 + 𝑦, 2𝑦 + 𝑧, 2𝑧 + 𝑤)
For the ones that are linear, find the matrix representation (in the standard basis).
Find the dimensions of the Domain and Co-Domain, and determine whether the
transformation is one-to-one or onto.
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