Calculate the system parameter matrices and a linear

Question

Question 1:

Given the two vectors v₁ = (2, 3) and v₂ = (-2,-3), find the following graphically and write the result:

V₁ - v₂, v₁v₂, x v₁, x v₂

Question 2:

Solve the following system using A-column-space and output space:
2x + y = 3
3x =4

Question 3:

Solve the following system using A-row-space and input space:
2x + 3y = 5
-x + 2y = 2
y = 2 + 0.5x

Question 4:

A linear transformation is applied to the image on the left and we got the image on the right. Find the applied linear transformation matrix.

Question 5:

A dynamic system is defined by the following set of equations:

x^.1(t) = Ax (t) + bu(t)

y^.(1) = cx (t-1)

Where X(t) = [x₁ x₂]^T ∈ R² is the system state vector, u(t) ∈ R is the system input vector, y(t) ∈ R is the system output vector, x^.(t) = dx(t)/dt ≅ x(t₂) - x(t₁)/(t₂ - t₁) is the first derivative of state vector, A ∈ R^2x2, and c ∈ R^1x2 are the system parameters matrices.

An experiment was done to collect data and the obtained data is

t	u	x1	x2	y
0	0	2	-1	1.5
1	1	1.8097	-0.6042	1.5076
2	1	2.5891	1.2183	3.1983
3	1	3.2944	2.3194	4.4541
4	1	3.9325	2.9846	5.4249
5	1	4.51	3.3866	6.2032
6	1	5.0324	3.6294	6.8471
7	1	5.5052	3.7761	7.3932
8	1	5.9329	3.8647	7.8653
9	1	6.32	3.9183	8.2791
10	1	6.6702	3.9506	8.6455

Calculate the system parameter matrices A ∈ R^2x2, b ∈ R^2x1 and c ∈ R^1x2 using the above information.