Guide on Linear Transformations in Linear Algebra
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Linear transformations
A linear transformation is a transformation
additive property:
for all .homogeneity property:
for all and .
Functions as linear transformation
Suppose
Solution. Since functions transform the input
For any value
This means that
We now check for the homogeneity property:
Since both criteria are satisfied,
Non-linear transformation
Consider the following transformation
Solution. Let's check for the homogeneity property:
Since
Linear transformations map the zero vector to the zero vector
If
Any transformation that does not satisfy this must be non-linear. Note that the converse is not true, that is,
Proof. If
Let's set
This completes the proof.
Showing that a transformation is non-linear
Consider the following two functions below. Are they linear transformations?
Solution. We can use theoremlink to easily check if
This means that
Let's now check the linearity of
Unlike
Since the homogeneity property is not satisfied, we conclude that
Linear transformation as matrix-vector product
Let
Where
Proof. Any vector
Let's apply a linear transformation
Since
Here, note the following:
the second-to-last step used theoremlink to rewrite the linear combination using a matrix-vector product.
is an matrix whose columns are the transformed standard basis vectors , , , .
This means we can convert any linear transformation to a matrix-vector product by applying the transformation to the standard basis vectors! This completes the proof.
Writing a transformation as a matrix product
Consider the linear transformation
Write the transformation in matrix-vector product form.
Solution. The standard basis vectors for
Applying transformation
By theoremlink, the linear transformation can be expressed as:
Geometric interpretation of matrix transformation
Geometrically, a transformation can be interpreted as an operation that changes the position of the input. For instance, consider the following grid:

Now, consider the following linear transformation:
By applying this transformation to all the data points over the grid lines, we end up with the following result:

Notice how the grid lines are still straight and parallel - this is guaranteed because
Transforming our grid lines will result in non-straight non-parallel lines:

Let's now see what happens to the standard basis vectors after a linear transformation:
Notice how the transformed vectors are actually just the columns of the transformation matrix! Let's define
Let's visualize the standard basis vectors before and after the transformation:
Standard basis vectors before | Standard basis vectors after |
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![]() | ![]() |
Now, suppose we focus on a single point:
Let's visualize this point:
Point before | Point after |
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![]() | ![]() |
Here, we can make the following observations:
before the transformation - we see that
can be represented using 2 and 1 .after the transformation - we see that
can be represented using 2 and 1 .
The key here is that the point
Let's also verify this mathematically. We can express
This tells us that
Let's now transform
This tells us
Finding the transformed standard basis vectors
Consider the linear transformation
Without doing any computation, derive the transformed standard basis vectors.
Solution. Since we are in