In this problem, we're going to take a look at a linear transformation.
In general, linear transformations can be very general operations or a certain criteria
that they need to satisfy.
In this particular example, we're going to look at the transformation Ax.
So the linear transformation will be described in terms of a matrix operating on a vector
x.
So we're going to consider this linear transform.
LT is the linear transform T. So just in terms of general notation T, this
transformation is going to take some vector in R3, and return a vector in R3.
And specifically for this problem, the transformation is defined as Ax, where the matrix A is defined
as this 3 by 3 matrix, 1, 0, 3, 4, 5, negative 1, and 0, 0, 2.
So when we talk about transformations, generally we might just write this down, T of x.
Hey, the linear transformation takes the vector x and returns some output vector T of x.
But for this particular example, we're going to go ahead and write down this concrete instance.
It is equal to A times x.
It turns out any linear transformation always has a matrix representation like this.
But we're going to go ahead and explicitly state what this specific matrix is for this
problem.
So here's our matrix A. And in part a of the problem, we're going to work with the vector
x, which is equal to a negative 1, 2, and 5.
And we are asked to compute what T of x is.
So that's almost trivial.
To compute T of x, we write down what T of x is equal to.
It's equal to Ax, which, for this particular example, is equal to the matrix A times the
vector x.
Vector x is a negative 1, 2, 5.
So if we write this out, we know how to take a vector-- or, I'm sorry, a matrix times a
vector.
Each column of the matrix gets weighted by the corresponding entry of the vector.
So this is going to be equal to the first column of a times the first entry of x plus
the second entry of x times the second column of A plus the third entry of x times the third
column of A. And then we know how to do scalar multiplication.
And negative 1 times that first vector just changes the sign of every entry.
So it turns into a negative 1 and negative 4, 0.
2 times the second vector is just 2 times every element.
So we have zeros in the first and third entries.
And 2 times 5 is 10 in the second.
And then 5 times a third turns into 15 and negative 5, 10.
And then this is just vector addition, so we just add all these element-wise.
So a negative 1 plus 0 plus 15 turns into 14.
And negative 4 plus 10 minus 5 turns into 1, because that's a negative 9 plus 10 is
1.
And in the very last row, 0 plus 0 plus 10 is 10.
So we have computed the transformation of the vector x when x is equal to a negative
1, 2, 5 for the specific case where our linear transformation is described by this matrix
vector product.
Let's do another example.
In this example, it's the same type of thing.
We're going to also be considering the transformation T of x where T of x is A of x, but in this
case, x is equal to the vector a, b, c.
So we're just replacing a concrete instance with this variable a, b, c vector.
So T of x is still equal to x, but now A is the same, and x is now a, b, c.
We're going to do the math the exact same way.
I'm going to have a times the first column plus b times the second column plus c times
the third column.
So I can go ahead and write that out.
This is just the definition of matrix vector product.
Each column of A gets weighted by the corresponding entry in the vector b.
And then I can go ahead and multiply this out.
a times that first column is a, 4a, 0.
This one is 0, 5b, 0.
And the final one is 3c, a negative c, 2c.
If I add all these up, do vector addition, I'll get a plus 3c.
I'll get 4a plus 5b minus c.
And the final row is 0 plus 0 plus 2c.
And this is my answer.
This is the transformation of the vector x when x is equal to a, b, c and when the particular
linear transformation we're looking at has a matrix representation given by the matrix
A. So we've completed this problem.
We've looked at the linear transformation of two different vectors under the same linear
transformation A of x.