Example:

The matrix $$ \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}$$ doubles the first coordinate of each point, stretching all shapes horizontally (by a factor of 2).

The matrix $$ \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}$$ doubles the first coordinate of each point, stretching all shapes horizontally (by a factor of 2).

Example:

The matrix $$ \begin{bmatrix} 2 & 0 & 0\\0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ also doubles the first coordinate of each point, and also stretches the space horizontally by a factor of 2.

The matrix $$ \begin{bmatrix} 2 & 0 & 0\\0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ also doubles the first coordinate of each point, and also stretches the space horizontally by a factor of 2.

Example:

The matrix $$ \begin{bmatrix} 1 & 0 & 1\\0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ will shear 3D space...

The matrix $$ \begin{bmatrix} 1 & 0 & 1\\0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ will shear 3D space...

The base stays where it was, but we pushed the top over.

You might have a parallelepiped with you right now!

If we transform a 2D shape, we use a \(2\times 2\) matrix.

$$ \begin{bmatrix} a&b\\c&d \end{bmatrix}$$

If we transform a 3D shape, we use a \(3\times 3\) matrix.

$$ \begin{bmatrix} a&b&c\\d&e&f\\g&h&i \end{bmatrix}$$

To translate a 2D, we need to add the same value to each \(x\) and \(y\) coordinate.

Example:

$$ \begin{bmatrix} x\\y \end{bmatrix} + \begin{bmatrix} 3\\0 \end{bmatrix} = \begin{bmatrix} x+3\\y \end{bmatrix}$$ will translate the plane 3 units to the right.

$$ \begin{bmatrix} x\\y \end{bmatrix} + \begin{bmatrix} 3\\0 \end{bmatrix} = \begin{bmatrix} x+3\\y \end{bmatrix}$$ will translate the plane 3 units to the right.

But all hope is not lost for the fans of linear transformations!

We can write a 2D translation as a matrix... we just need an extra dimension.

We can write a 2D translation as a matrix... we just need an extra dimension.

The only problem is, we can't multiply this matrix by a vector with only two components, like \(\begin{bmatrix} x\\y \end{bmatrix}\).

$$ \begin{bmatrix} x\\y\\1 \end{bmatrix}$$

$$ \begin{bmatrix} 1 & 0 & a\\0 & 1 & b\\0 & 0 & 1 \end{bmatrix}\begin{bmatrix} x\\y\\1 \end{bmatrix} = \begin{bmatrix} x+a\\y+b\\1 \end{bmatrix}$$

Example:

$$ \begin{bmatrix} 1 & 0 & 3\\0 & 1 & 1\\0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x\\y\\1 \end{bmatrix} = \begin{bmatrix} x+3\\y+1\\1 \end{bmatrix}$$ will move every point 3 units to the right and 1 unit up.

$$ \begin{bmatrix} 1 & 0 & 3\\0 & 1 & 1\\0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x\\y\\1 \end{bmatrix} = \begin{bmatrix} x+3\\y+1\\1 \end{bmatrix}$$ will move every point 3 units to the right and 1 unit up.

As you might guess, we'll need a \(4 \times 4\) matrix.

$$ \begin{bmatrix} 1 & 0 & 0 & a\\0 & 1 & 0 & b\\0 & 0 & 1 & c \\ 0 & 0 & 0 &1 \end{bmatrix}$$

The math works the same way it always did:

$$ \begin{bmatrix} 1 & 0 & 0 & 2\\0 & 1 & 0 & -3\\0 & 0 & 1 & 5 \\ 0 & 0 & 0 &1 \end{bmatrix}\begin{bmatrix} x\\y\\z\\1 \end{bmatrix}=\begin{bmatrix} x+2\\y-3\\z+5\\1 \end{bmatrix}$$

Which begs the question: just what is 4D space?

They were mainly used to solve systems of linear equations (their big use in linear algebra classes today).

The bracket notation we use now didn't come along until 1913, and took awhile to be adopted.

In fact, this is how complex numbers started being taken seriously by mathematicians!

To see how this worked, think of a point \((x,y)\) as a complex number, \(x+iy\). If we multiply this number by \(i\), we get $$i(x+iy) = xi + i^2y = xi-y = -y + ix$$ which we can think of as the point \((-y,x)\).

Example:

The point \(P = (2,3)\) becomes the complex number \(2+3i\). Multiplying by \(i\) we get $$ i(2+3i) = 2i + 3i^2 = 2i - 3 = -3 + 2i$$ which is the point \(P' = (-3,2)\).

The point \(P = (2,3)\) becomes the complex number \(2+3i\). Multiplying by \(i\) we get $$ i(2+3i) = 2i + 3i^2 = 2i - 3 = -3 + 2i$$ which is the point \(P' = (-3,2)\).

Example:

Similarly, the point \(Q = (3,2)\) becomes the complex number \(3+2i\). Multiplying by \(i\) we get $$ i(3+2i) = 3i + 2i^2 = 3i - 2 = -2 + 3i$$ which is the point \(Q' = (-2,3)\).

Similarly, the point \(Q = (3,2)\) becomes the complex number \(3+2i\). Multiplying by \(i\) we get $$ i(3+2i) = 3i + 2i^2 = 3i - 2 = -2 + 3i$$ which is the point \(Q' = (-2,3)\).

This makes sense, because we only need a two dimensional matrix to rotate.

Rotations in 2D using complex numbers was pretty well understood in the 1800s.

Sir William Rowan Hamilton (1805 – 1865) was an Irish mathematician that tried to find a similar way of rotating 3D shapes.

Sir William Rowan Hamilton (1805 – 1865) was an Irish mathematician that tried to find a similar way of rotating 3D shapes.

He tried adding another imaginary component, something like
$$ a + bi + xj $$
but the math never worked out consistently.

He was about to give up, when he went for a walk...

He was about to give up, when he went for a walk...

While crossing a bridge in 1843, he had a revelation: he needed 3 imaginary numbers to make it work.

He got so excited he carved his idea right into the bridge. His "note to self" is worn away, but this plaque is still there to commemorate the moment.

He got so excited he carved his idea right into the bridge. His "note to self" is worn away, but this plaque is still there to commemorate the moment.

$$ a + bi + cj + dk$$

We won't worry much about the meaning of this number, but \(i\), \(j\) and \(k\) are all "imaginary".

The numbers of this form are called

$$ a + bi + cj + dk$$

has four parameters: \(a\), \(b\), \(c\), and \(d\).

It's a four dimensional number!

There must be a meaning behind them, but what?

The three axes we know about are perpendicular, there's no room for another one!

Those are our three dimensions.

(Note: not to scale, we actually can't see a point).

That's a nice cube!

To make a 1D cube, make two copies of the 0D cube, and connect with a line:

We usually call this a square!

At least we see the cube now.

In mathematics, this is referred to as a tangled mess.

To begin to understand pictures of it, we need to remember that we're only seeing a

Even a 3D model of a 4D cube isn't accurate, and a 2D model is worse.

The lines that seem to "cross" here don't really go near each other!

That might not help much, but it's a start.

There's something important about this that's easy to miss. The edges of the cube that meet at any corner are all perpendicular.

Not only that, but each of the "inner cells" are all the same size - and they're all 3D cubes. 7 of them, to be exact.

The big cube surrounding them all? It's the same size as the little cube inside!

Does that mean it doesn't exist?

We can't even see an actual circle, but they exist!

It even has an entire Wikipedia page.

By the way, a tesseract can also be called a

The idea of the fourth dimension generated a lot of interest around 1900. We'll look at the different theories on what it is and how to think about it, and how these theories were embraced by artists.

Until then, here's Salvador Dali holding his beloved tesseract.

Until then, here's Salvador Dali holding his beloved tesseract.