Need help with following MatLab Problems 

 

close all;

clear all;

 

% This is a script. It is a useful way to write many MATLAB commands and

% then run them all at once. Later we will also use scripts to write

% functions, creating tools that we could use later in future scripts

%

% In class we saw how to:

% -Create variables

% -Create vectors

% -Access entries of vectors

% -Add vectors

% -Multiply vectors by scalars

% -Plot points and vectors

% -Write simple for loops

%

% Here we will explore linear combinations of vectors, with a particular

% focus on the number of dimensions a fixed set of given vectors can 'span'.

 

 

 

%% Linear combination of two vectors in 2D #1

% Here we want to consider different linear combinations of two vectors in

% 2D. Our vectors will be v1 = [1,2] and v2 = [4,-1]. Produce a plot

% showing the original vectors using the quiver command and 100 different

% random linear combinations of those two vectors using plot.

 

v1 = [1,2];

v2 = [4,-1];

 

figure();

title('The linear combinations of 2 vectors');

hold on;

 

for i=1:100

c = 2*rand(1,2)-1;

x = c(1).*v1 + c(2).*v2;

plot(x(1),x(2),'ro')

end

 

quiver(0,0,v1(1),v1(2),'b','linewidth',2);

quiver(0,0,v2(1),v2(2),'k','linewidth',2);

 

%% Questions

% #1: Can any vector in the plane be expressed as a linear combination of

% of v1 and v2? Why do you think that?

%

% Answer: I do think that any vector in the plane can be expressed as a

% linear combination since v1 and v2 are linearly independent of eachother

%

% #2: How would the plot look different if the coefficients c were sampled

% differently? For instance, what happens if c = 10*rand(1,2)

%

% Answer:

 

%% Linear combination of two vectors in 2D #2

% Here we want to consider different linear combinations of two vectors in

% 2D. Our vectors will be v1 = [1,2] and v2 = [-2,-4]. Produce a plot

% showing the original vectors using the quiver command and 100 different

% random linear combinations of those two vectors using plot.

 

v1 = [1,2];

v2 = [-2,-4];

 

figure();

title('The linear combinations of 2 vectors');

hold on;

 

for i=1:100

c = 2*rand(1,2)-1;

x = c(1).*v1 + c(2).*v2;

plot(x(1),x(2),'ro')

end

 

quiver(0,0,v1(1),v1(2),'b','linewidth',2);

quiver(0,0,v2(1),v2(2),'k','linewidth',2);

 

%% Questions

% #3: Can any vector in the plane be expressed as a linear combination of

% of v1 and v2? Why do you think that?

%

% Answer:

%

% #4: Looking back at v1 and v2, what about those vectors explains why this

% is differrent than the first set of v1 and v2.

%

% Answer:

%

% #5: Why do the red points extend beyond both v1 and v2?

%

% Answer:

 

%% Linear combination of two vectors in 3D #1

% Here we want to consider different linear combinations of two vectors in

% 3D. Our vectors will be v1 = [1,2,0], and v2 = [4,-1,2]. Produce a plot

% showing the original vectors using the quiver3 command and 100 different

% random linear combinations of those two vectors using plot3.

 

v1 = [1,2,0];

v2 = [4,-1,2];

 

figure();

title('The linear combinations of 2 vectors in 3D');

hold on;

 

for i=1:100

c = 2*rand(1,2)-1;

x = c(1).*v1 + c(2).*v2;

plot3(x(1),x(2),x(3),'ro')

end

 

quiver3(0,0,0,v1(1),v1(2),v1(3),'b','linewidth',2);

quiver3(0,0,0,v2(1),v2(2),v2(3),'k','linewidth',2);

 

view(3)

 

%% Questions

% #6: What vectors in 3-spaces can be expressed as a linear combination of

% of v1 and v2? Why do you think that, and could you predict that from the

% problem setup?

%

% Answer:

 

 

%% Linear combination of three vectors in 3D #2

% Here we want to consider different linear combinations of three vectors in

% 3D. Our vectors will be v1 = [1,2,0], v2 = [4,-1,0] and v3 = [0,0,1]. Produce a plot

% showing the original vectors using the quiver3 command and 100 different

% random linear combinations of those two vectors using plot3.

 

%Please write code

 

%% Questions

% #7: Can all vectors in 3-space be expressed as a linear combination of

% of v1, v2 and v3? Why do you think that and could you predict that from

% the problem setup?

%

% Answer:

 

 

%% Linear combination of three vectors in 3D #3

% Here we want to consider different linear combinations of three vectors in

% 3D. Our vectors will be v1 = [1,2,0], v2 = [4,-1,2] and v3 = [-5,-1,-2]. Produce a plot

% showing the original vectors using the quiver3 command and 100 different

% random linear combinations of those two vectors using plot3.

 

%Please write code

 

%% Questions

% #8: Can any vector in 3-space be expressed as a linear combination of

% of v1, v2 and v3?

%

% Answer:

%

% #9: Why is your answer to #8 different to your answer to #7 and could you

% tell from equations for v1, v2 and v3 that this would be the outcome?

%

% Answer: