Some matrix vector product.

So turn this from a vector, vector dot product to a matrix, matrix product.

Now, can we take this matrix vector product?

But when you take this product, this matrix vector product, what do you get?

You take the dot product of this column, this row vector with this column vector.

All linear transformations can be a matrix vector product.

So when we distribute the matrix vector product, you get

But we also know that any matrix product, any matrix vector product, is also a linear transformation.

So the dot product of this row vector with this column vector should be equal to that 0.

So this is essentially the dot product of this row vector.

But the dot product of two vectors is equal to the product of their lengths, their vector lengths.

We're taking each row and we're essentially taking the dot product of this row vector with this column vector.

We can now rewrite this transformation here as the product of any vector.

So that's going to be the first entry in this matrix vector product.

And any linear transformation you could actually represent as a matrix vector product.

And we can actually take the dot product of this row vector and this column vector because they have the same length.

Well my matrix times vector product, or multiplication, is only defined if my vector has as many components as this matrix has columns.

And then the fourth column in our product vector is going to be the matrix A times the column vector 1, minus 1, 2.

Well, now we've reduced our matrix matrix product problem to just four different matrix vector product problems, so we can just multiply these.

But anyway, back to our attempt to represent this transformation as a matrix vector product.

So let me find a normal vector to them both by taking the cross product.

I got this result by figuring out the basis of the column span, finding a normal vector by taking the cross product of our two basis vectors, and then using the dot product of the normal vector with the difference this vector right here, where you take any vector on our plane minus one of our basis vectors, that's to find some vector in the plane.

Now, I just told you that I can represent this transformation as a matrix vector product.

We defined it as, the projection of x onto L was equal to the dot product of x, with this defining vector. x dot this defining vector, divided by that defining vector dotted with itself.

Well, we're still on the first row of the product vector but now we're on the second column.

We've seen in a previous video that any linear transformation can be represented as a matrix vector product.

We saw that a long time ago, that any linear transformation could be represented as a matrix vector product.

We have more columns here than entries here, so we have never defined a matrix vector product like this.

The first term of this is going to be the dot product of this first row with this vector.

And then the second entry is going to be the dot product of this row vector with this column.

So AB, let me rewrite it AB, my product vector, is going to be equal to so this first column is the matrix A times the column vector 1, 2, 3.

So we now know that our normal vector 5, minus 1, minus 1, that I got by taking the cross product of our basis vectors dot any vector in our plane.

In the next video I'm going to show you that any linear transformation this is incredibly powerful can be represented by a matrix product or by any transformation on any vector can be equivalently, I guess, written as a product of that vector with a matrix.

We can now write our transformation. Our transformation of x1, x2 can now be rewritten as the product of this vector.

We see it in our dot product properties, this is equal to c times a dot u, times the vector u.

I would have just taken the dot product of the reduced row echelon form or, not the dot product, the matrix vector product of the reduced row echelon form of A with this vector, and I would've gotten these equations, and then these equations would immediately, I can just rewrite them in this form, and I would have gotten our result.

So we could rewrite it vector 1, vector 2, vector 3, and vector 4.

Vector a is equal to vector b plus vector a minus b.

And if you have that constraint, if the length of your vector, or the number of components in vector is equal to the number of columns in your matrix, then we define this product to be equal to so this is my vector x so this is a definition.

Well, it's this vector plus that vector.

I multiply this vector times this vector.

length of vector a minus vector b.

My first basis vector was the vector 2 minus 1 and my second basis vector is the vector 2, 1.

The scalar product also allows one to define an angle between two vectors formula_30In formula_31 dimensional Euclidean space, the standard scalar product (called the dot product) is given by formula_32 Cross product in 3 dimensional space The cross product of two vectors in 3 dimensions is a vector perpendicular to the two factors, with length equal to the area of the parallelogram spanned by the two factors.

I'm going to multiply that vector times that row vector times this column vector.