1. Linear Transformations
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2. 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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3. 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
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.
A few examples follow…
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4. Linear Transformations
This matrix is 2x2, so it transforms vectors from ℝ2 into (other) vectors from ℝ2.
To see what this matrix does, we can see where it takes a few specific vectors.
Domain (ℝ2)
Range (ℝ2)
𝑥 → 𝐴 1 𝑥
This is a shear transformation.
Notice also that the columns of the matrix are the images of the Standard Basis vectors.
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5. Linear Transformations
This matrix is 2x2, so it transforms vectors from ℝ2 into (other) vectors from ℝ2.
To see what this matrix does, we can see where it takes a few specific vectors.
Domain (ℝ2)
Range (ℝ2)
𝑥 → 𝐴 2 𝑥
This matrix is a combination of a rotation through 45° and a stretch by a factor of √2.
We may have more to say about this type of matrix in a few weeks.
Check out your textbook for more discussion of 2x2 transformation matrices.
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6. Linear Transformations
𝐴 3 = −1 3
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.
Domain (ℝ2)
Range (plane in ℝ3)
𝑥 → 𝐴 3 𝑥
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 the Range is not all of ℝ3.
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7. Linear Transformations
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
Range is
All of ℝm T
Domain ℝn
Range is a proper subspace of ℝm T
T is onto
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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8. Linear Transformations
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.
This happens when there is a pivot in every row.
A transformation is one-to-one iff the columns are linearly independent.
This happens when there is a pivot in every column.
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9. Linear Transformations
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.
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
(check this yourself!)
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10. 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
𝑇 𝑐 𝑢 =𝑐𝑇 𝑢
For all scalars c and every vector u in the domain of T
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.
𝑇 0 = 0
An important special case of the scalar multiplication rule is that
This gives an easy way to test a transformation for linearity.
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11. Linear Transformations
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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12. Linear Transformations
𝑎) 𝑇 𝑥,𝑦 =(2𝑥,𝑥+𝑦)
This is a linear transformation (from ℝ 2 to ℝ 2 ).
The formulas involve only linear combinations of the inputs.
In order to prove that it is linear, use the definition:
𝑇 𝑥 1 + 𝑥 2 , 𝑦 1 + 𝑦 2 =(2( 𝑥 1 + 𝑥 2 ), (𝑥 1 + 𝑥 2 )+( 𝑦 1 + 𝑦 2 ))
=(2 𝑥 1 , 𝑥 1 + 𝑦 1 )+(2 𝑥 2 , 𝑥 2 + 𝑦 2 )=𝑇( 𝑥 1 , 𝑦 1 )+𝑇( 𝑥 2 , 𝑦 2 )
𝑇 𝑐𝑥,𝑐𝑦 =(2𝑐𝑥,𝑐𝑥+𝑐𝑦)=𝑐𝑇 𝑥,𝑦
To find the matrix for the transformation, see what happens to the standard basis vectors:
𝑇 = 2 1
𝑇 = 0 1
The images of the standard basis vectors are the columns of the transformation matrix. 𝐴=
𝑇 𝑥 𝑦 = 𝑥 𝑦 = 2𝑥 𝑥+𝑦 √
The domain and range are both ℝ 2 .
The matrix has a pivot in every row and every column, so it is one-to-one and onto.
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13. Linear Transformations
𝑏) 𝑇 𝑥,𝑦 =(𝑥−3𝑦,𝑥𝑦)
This one is a transformation from ℝ 2 to ℝ 2 , but is not linear.
The second component involves a nonlinear combination of the inputs.
We can prove that this is not linear by showing an example where the definition is not satisfied:
For example, 𝑇 = −1 2 , but 𝑇 = −2 8 .
We expect to get −2 4 (twice the input = twice the output). This contradiction shows the transformation is not linear.
I chose the vector at random.
We could show this contradiction in general by comparing
𝑇 𝑐𝑥 𝑐𝑦 = 𝑐𝑥−3𝑐𝑦 𝑐 2 𝑥𝑦 and c𝑇 𝑥 𝑦 = 𝑐𝑥−3𝑐𝑦 𝑐𝑥𝑦 . Not equal, so not linear.
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14. Linear Transformations
𝑐) 𝑇 𝑥,𝑦,𝑧 =(𝑥,𝑦,0)
This one is a linear transformation from ℝ 3 to ℝ 3 .
To find the matrix, use the standard basis vectors:
𝑇 =
𝑇 =
𝑇 = 𝐴=
Notice this is already in reduced echelon form. There are pivots in the first two columns, and the first two rows.
This transformation is neither one-to-one or onto.
The domain is all of ℝ 3 .
The range is a 2-D subspace of ℝ 3 , namely the x-y plane.
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15. Linear Transformations
𝑑) 𝑇 𝑥,𝑦,𝑧 =(2𝑥,2𝑦,2)
This one is a transformation from ℝ 3 to ℝ 3 , but is not linear.
We can just check the zero vector on this one.
𝑇 =
Since 𝑇 0 ≠ 0 this transformation is not linear.
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16. Linear Transformations
This one is a linear transformation from ℝ 4 to ℝ 3 .
Again, the matrix is found from the standard basis vectors. 𝐴=
This has a pivot in every row, but not every column.
So this transformation is onto, but not one-to-one.
Domain is all of ℝ 4 .
Range is all of ℝ 3 .
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