Suppose we have a number.
By changing the value of this one number,
we can represent any point in a one dimensional space.
Now suppose we have two numbers.
By changing the values of these two numbers,
we can represent any point in a two dimensional space.
Now suppose we have three numbers.
We can now represent any point in a three dimensional space.
Similarly, if we have a set of four numbers,
we can represent any point in a four dimensional space.
If we have a set of five numbers,
we can represent a point in five dimensional space.
And if we have an infinite set of numbers,
we can represent a point in a space with an infinite number of dimensions.
Regardless of how many dimensions we have,
each set of numbers which represents a point in space
can be thought of as an arrow, which we will call a vector.
Each vector has a length, and a direction.
If we multiply the vector by a number, the vector’s length will change,
but its direction will stay the same.
If we add two vectors together, the result is as shown.
Each vector can be thought of as the
sum of vectors pointed in each of the different dimensions.
And each of the vectors pointed in each of the dimensions can be thought of as
a vector a length of one multiplied by a number.
There are many transformation that can be applied to a vector.
For example, the vector can be rotated in space around this axis,
by a certain number of degrees.
Suppose we apply this rotation to a group of vectors.
And then we add all the vectors together.
Now, suppose that instead of starting out by rotating the vectors,
we instead first add all of all the vectors together,
and then afterwards we rotate the result.
It turns out that the final vector that is produced is the same as before.
If the final vector that is produced is the same regardless of
whether the vectors are added together before or after the transformation,
then we say that this is a linear transformation.
Another example of a linear transformation is having
the vector stretched or squeezed as shown.
Or having the vector flipped into this mirror image of itself.
Or having the vector stretched in this dimension,
and then flipped.
In each of these cases, the final vector that is produced is the same regardless
of whether the vectors are added together before or after the transformation.
In each of these cases, the final vector that is produced is the same regardless
of whether the vectors are added together before or after the transformation.
This is the definition of a linear transformation.
We can represent a linear transformation mathematically by showing
how the transformation would affect vectors with a length of one,
pointed in each of the dimensions.
The original vectors are vectors with a length of one,
pointed in each of the dimensions.
After the transformation is applied, the new vectors are the result.
Suppose we take each of the new vectors that result from this,
and we line up all their numbers together as is shown.
We now have what we call a matrix.
All linear transformations can be represented with a matrix.
When we apply the transformation, we say that we are
multiplying the vector by the matrix, and we represent it like this.
We defined the numbers in each column of the matrix based on the result of
the transformation of vectors of length one in each of the dimensions.
Therefore, when the matrix is multiplied by a vector of length one
in one of the dimensions, the result is a follows.
If we want to find out how the transformation affects a vector pointed
in one of the dimensions that is not length one, then we can do the following.
And if we want to find out how the transformation affects any vector
of any length pointed in any direction, then we can do this as shown.
This is how a matrix is multiplied by a vector,
and it is how all linear transformations can be computed.
This is true for four dimensions
For Five dimensions
And for an infinite number of dimensions
Much more information about mathematics
is available in the other videos on this channel.
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