Stage = structure prototype(comple)

What’s a coordinate

emember how in how linear combination thing we learn that a how to reach a vector on the plane? How we created a linear combination of a line by multiplying it with scalars. We can see the coordinate using the same idea only that in our system we use the standard basis vectors

In 2D space, we saw from article x that we can combine two vector to reach another vector. We saw that by combining two vectors of different direction we can reach any other vector in the 2D space. How significant is this fact?

Basis vectors represents the whole space

The whole 2D plane can be represent by a combination of just two simple vectors. It means that any other coordinate in a space can be defined as a combination of those vectors. We call those foundational vectors, the basis vector. It creates a new way to think of a coordinate. If our basis vector are (1,0) and (0,1). Any coordinate a coordinate like (5,3) can be seen as

Linear combination equation We strech (1,0) by five and and a streched (0,1) on top.

Try coordinate in a system with no basis vector

What if we choose the basis vector to be () and (). We said that a coordinate (5,3). Means that we scale the first basis by five then the second basis by three: (algebraic expression that shows this)

Whereas in the previous space with basis (1,0) (0,1), you reached (5,3) after the combination, in this space you reach the vector (,)

What’s the difference between those 2 planes. The first one had what we call unit vector as basis. When you combine unit basis according to a coordinate, you the vector at the end of that combination is the same as your starting coordinate. If the vectors aren’t unit, then you get. If i say move 5 steps forward and each step is one 1 ft you end up at 5 ft but if each steps is not a unit step like 4 you end up at 20 after moving 5 steps.

Also note how this process of multiplying ressemble matrix multiplication that will be important later on

Can you see the parallel between matrix multiplication(Here you link your article on linear combination) of a matrix and a vector and our our definition of linear combination of a coordinate with basis vector? This will be the most important piece soon?

A transformation changes maps every vector in a space into another a corresponding vector

A transformation in mathematics means a function. A way to map elements of a set with another one. Transformation crops up in linear algebra. Let’s say we take the space formed our 2D space formed from our basis (1,0) and (0,1). Imagine a function that will multiply every vector in 2D cartesian plane by 2. (1,0) becomes. Coordinate (x,y) becomes . A transformation on a space is a function whose input are every vector of a space. It transforms that space every vector of that space into a new vector so that we get a completely new space.

If a transformation is linear it has three properties concerned with addition and multiplication

A Linear transformation in addition to mapping those vectors has two following conditions: (give those conditions in latex)

if you act out visually what linear transformation does you see that it preserves multiplication and addition exactly as we know it

Visually, we see that a + b = vector c . Then if we tranformed c to give c transformed. We can reach that same transformed, if we directly did that linear combination with a tr and b tr that will give c tr. Addition in vector space one correspond to exactly to addition in vector space 2. We say the property of addition is preserved in a linear transformation

Same demonstration for scalar multiplication

multiplication is preserved

There’s an isomorphism between those two spaces and their operations

A linear combination you take here corresponds exactly to a linear combination in the transfomed system. (precise that we call it isomophic)

Remember how we say that a whole space can be formed with just just addition and muliplication of basis vectors? The idea is instead of asking how each individual vectors will be ttransformed let’s just make sure we know how those basis vector are transformed and since we preserve addition and muliplication, we will be able to reconstruct the new space after the transformation

Remember that a linear combination of basis vector defines the whole space. A linear combination is made up of addition and multiplication. Since addition and muliplication are preserved after a linear transformation. Instead of representing the transformation for all vectors of our original space. Just tracks what the new transformed base. From them you will reconstruct the new transformed space of vector. Look again at the properties of linear transformation and how the preserve operations.

So the whole linear tranformation can be placed into a collection of vectors of the two transformed vectors: (place a matrix here)

That’s what a matrix is a combination of those new vectors

When you take a coordinate in your new system. For example (,) you combine them with those basis vector. To give

When you say Ax=b. if you remember what we say of coordinate is that theyare the same thing in different dependding on the basis

matrix multiplication is just finding what those coordinate represent in the new transformed state

If you track the transformed basis vector, you can reconstruct the transformed vector space this is because we assume that a linear tranformation preserves those fundamental operations.