Show given vector is in the span of two other vectors

Hello, so i was given the question;

Assume that a given vector u is in the span of vectors v and w in Rn. This means that there exist scalars a,b such that av+bw=u. I was given a matrix M of size 3×3. The first row of M is the vector v, the second row is the vector w, and the third row is the vector u.

1,2,0

1,3,5

3,8,10

I assume for this to work the equation * would be a[1,2,0] + b[1,3,5] = [3,8,10]

and thus separated the vectors v = M(1,:); w = M(2,:); u = M(3,:);

I was told to use x = linsolve(A, B) to solve *

however I am unsure how to assign and what to assign A and B (I've tried A = [a*v+b*w] but that came up as an error)

Thanks!

Answer 1

[Divyam]

Hi @amy wang,

Using the "linsolve" function to set up the system of linear equations indeed is the solution to your problem. You just need to assign the matrices on the LHS of the equation to the matrix A and the RHS to matrix B.

The "linsolve" function will then model this system of linear equations in the form, and solve this form to obtain the coefficients, a and b. Here is a sample code to help you with the same:

% Define the matrix M
M = [1, 2, 0; 1, 3, 5; 3, 8, 10];

% Extract vectors v, w, u
v = M(1, :)';
w = M(2, :)';
u = M(3, :)';

% Construct the matrix A using v and w
A = [v, w];

% Vector B is u
B = u;

% Solve for x = [a; b] using linsolve
x = linsolve(A, B);

% Display the results
a = x(1);
b = x(2);
fprintf('The scalars are a = %.2f and b = %.2f\n', a, b);

The scalars are a = 1.00 and b = 2.00

For more information regarding how you can use the "linsolve" function to solve a system of linear equations, you can look at the examples listed in this documentation: https://www.mathworks.com/help/matlab/ref/linsolve.html