Solve for matrix in linear equation

You may search for a rotation matrix that turns the direction of u into the direction of v. Applying this matrix to u gives a vector in the direction of v, but with a different length. Therefore you must adjust the rotation matrix by multiplying by the length of v and divide by the length of u. THis can be done in MMA by:

{u, v} = RandomReal[{-1, 1}, {2, 3}]
m = RotationMatrix[{u, v}]  Norm[v]/Norm[u]

m maps now u into v:

m.u == v
(*True*)

There are too many matrixs satisfy the equation M.u==v.

Without loss general, we can Normalize the vector u and v.

Besides the RotationMatrix which have mention by @Daniel Huber, we can construct the reflect-matrix by hand which also satisfiy the equation M.u==v

u = Normalize[{x, y, z}, Sqrt[#.#] &];
v = Normalize[{a, b, c}, Sqrt[#.#] &];
normal = Normalize[u - v, Sqrt[#.#] &];
M = IdentityMatrix[3] - 2 Outer[Times, normal, normal];
M.u == v // Simplify

True

ReflectionMatrix maybe another choise.

u = Normalize[{x, y, z}, Sqrt[#.#] &];
v = Normalize[{a, b, c}, Sqrt[#.#] &];
normal = Normalize[u - v, Sqrt[#.#] &];
M = Simplify[ReflectionMatrix[normal], normal ∈ Reals];
M.u == v // Simplify

True

For difference dimensions vector u and v,we can also combine the matrixs such as projection matirx and reflection matrix and rotation matirx to construct such matrix.

You can use a fitting procedure to find a solution. It's not going to be unique, though.

u = RandomReal[1, {3}]
v = RandomReal[1, {5}]
mat = Array[x, {Length[v], Length[u]}]
sol = NMinimize[Total[(mat.u - v)^2], Flatten[mat]];
mat /. Last[sol]

Check that the residuals are 0:

mat.u - v /. Last[sol]