Hello it's a am again. Today we continue with **Linear Algebra** getting into **Linear Functions Matrixes **and also some **special cases**. Last time we covered the basics of Linear Functions. So, without further do let's get started!

## Linear Function Matrix:

Suppose two vector spaces V and W with Bv = {v1, v2, ..., vn} being a basis of V and Bw = {w1, w2, ..., wm} a basis of W. As we said last time, we can construct a linear function f: V -> W, determining that f(vi) (1<=i<=n) are the images of the basis vectors of Bv.

Then the image of every vector *x = k1*v1 + k2*v2 + ... + kn*vn* of V, where ki are real is defined as:

**f(x) = k1*f(v1) + k2*f(v2) + ... + kn*f(vn)**

The images of f(vi) (1<=i<=n) as vector of W can be written as linear combinations of the basis vectors of Bw.

Let's say those are:

*f(v1) = a11*w1 + a21*w2 + ... + am1*wm*

*f(v2) = a12*w2 +a22*w2 + ... + am2*wm*

*..................................................................*

*f(vn) = a1n*w1 + a2n*w2 + ... + amn*wm*

where aij is a real number and 1<=i<=m and 1<=j<=n

That way we can** construct a matrix using the coefficients of aij** that is called the **linear function matrix **or **linear morphism matrix**.

We get these coefficients from the bases Bv and Bw of the vector spaces V and W.

So, we can define this **function **also using:

*f(v) = A*v*, for every v in V

### Example:

Suppose a linear morphism f: R^3 -> R^3, where f(x, y, z) = (x + 2y + z, x + 5y, z).

We will use the standard basis {e1, e2, e3} of R^3, where e1 = (1, 0, 0), e2 = (0, 1, 0) and e3 = (0, 0, 1).

That way we have:

f(e1) = (1, 1, 0) = 1*e1 + 1*e2 + 0*e3

f(e2) = (2, 5, 0) = 2*e1 + 5*e2 + 0*e3

f(e3) = (1, 0, 1) = 1*e1 + 0*e2 + 1*e3

So, the function matrix is:

1 2 1

1 5 0

0 0 1

### Transition Matrix:

The square matrix that we get from the indicator function Iv: V -> V from the bases B1 = {v1, v2, ..., vn} and B2 = {u1, u2, ...un} is called a **transition matrix** from basis B1 to B2. We write it as PB1->B2.

That way the transition matrix has the** coordinates of the vectors of basis B1 when in basis B2**.

This means that:

*[v1 v2 ... vn] = [u1 u2 ... un]*PB1->B2*

In the same way we can get:

*[u1 u2 ... un] = [v1 v2 ... vn]*PB2->B1*

### Invertibility:

For any two bases B1, B2 of V we know that PB2->B1*PB1->B2 = In (indicator matrix)

So, PB1->B2 = PB2->B1^-1 (inverse) and so the transition matrix is **invertible**.

We can also say tha same for the other way around:

PB2->B1 = PB1->2^-1 (inverse)

### Isomorphism Theorem:

Suppose V, W are two vectors spaces and Bv = {v1, v2, ..., vn} is a basis of V and Bw = {w1, w2, ..., wm} a basis of W. If f is a function in the homomorphism Hom(V, W) and A is a the morphism matrix that we get from f when going from Bv to Bw then:

Hom(V, W) -> M mxn, transitioning using homomorphism h having f->A using the same h.

The function Hom(V, W) -> M mxn is a **isomorphism**.

When V and W are vector spaces and **dimV = n, dimW = m** then **dimHom(V, W) = m*n**.

### When A is the mxn function matrix of f: V->W on the bases Bv and Bw then:

- N(A) = Kerf
- dimImf = rank(A)

When the the function f is also a homomorphism (f: V->V) we know that the matrix A^-1 (inverse matrix of A) is correspnding to the function f^-1 that is the inverse function of f.

### Equality and Similarity:

Two mxn matrixes A, B are **equivalent **when there are two invertible matrixes P in M^m and S in M^n so that:

*B = P^-1*A*S*

Two square matrixes A, B are **similar **when there is a invertible matrix P in M^n so that:

*B = P^-1*A*P*

When f: V->V is a homomorphism of V and A is the function matrix with basis B = {v1, v2, ..., vn} and B is a function matrix of basis B' = {v1', v2', ..., vn'} then A and B are similar.

## Special Cases:

A function that is of the form f: R^2 -> R^2, where f(x, y) = (ax, ay) and a in R is called a **expansion **when a>=1 and a **compression **when 0<a<1.

The function matrix can be represented as:

a 0

0 b

When a = 1: we move in the y'y axis

When b = 1: we move in the x'x axis

We can also define a** rotation** function where (x, y) ->(xcosφ + ysinφ, -xsinφ + ycosφ)

That way the function matrix looks like this:

cosφ -sinφ

sinφ cosφ

We can also define the **projection **Pr of a line with angle θ and the matrix looks like this:

cos^2θ cosθ*sinθ

cosθ*sinθ sin^θ

And this is actually it for today and I hope you enjoyed it!

Next time in Linear Algebra we will get into eigenvalues and eigenvectors.

Bye!