(Maple) Linear Combination of Matrices

Question

Problem: [This problem is intended to be done with Maple] Suppose that:

$$\textbf{u}_1=\begin{pmatrix} 356\\ -185\\ -580\\ -918\\ 147\\ 468\\ 504\\ 594 \end{pmatrix}, \textbf{u}_2=\begin{pmatrix} 573\\ 230\\ -950\\ -877\\ 69\\ 677\\ 323\\ 486 \end{pmatrix}, \textbf{u}_3=\begin{pmatrix} -1000\\ -833\\ 552\\ 935\\ -999\\ -483\\ 450\\ 457 \end{pmatrix}, \textbf{u}_4=\begin{pmatrix} 946\\ -379\\ -971\\ -840\\ 979\\ -591\\ -139\\ -451 \end{pmatrix}$$

Also suppose that $A = (\textbf{u}_1|\textbf{u}_2|\textbf{u}_3|\textbf{u}_4)$. Let $\textbf{v}=\begin{pmatrix}-27\\59\\-75\\21\end{pmatrix}$. Hence let $A\textbf{v}$ is a linear combination in the form

$$\lambda_1\textbf{u}_1 + \lambda_2\textbf{u}_2 + \lambda_3\textbf{u}_3 + \lambda_4\textbf{u}_4$$

What are the values of $\lambda_1, \lambda_2, \lambda_3$ and $\lambda_4$.

My attempts

If I'm being honest, I've been messing with the LinearAlegbra package for several hours and I have no idea if I'm on the right track. The first thing I did was use Gaussian Elimination on $A$ (which I defined as $B$), then used LinearSolve($B$,$\textbf{v}$) to find the values, although I do not believe this is correct (or at least the times I've tried always resulted in incorrect values).

Any help would be greatly appreciated!

Answer 1

A := Matrix([u1, u2, u3, u4]);

N := Matrix([A, A.v]);

ReducedRowEchelonForm(N);

Answer 2

You shouldn't need Maple for this at all; I can't see how it would be useful except to verify. There is also no reason to use Gaussian Elimination to reduce the matrix.

You just need the definition of $A\mathbf{v}$; this is commonly defined as a linear combination of the columns of $A$, using the entries of $\mathbf{v}$ as weights. So $\lambda_{1} = -27$, $\lambda_{2}=59$, $\lambda_{3}=-75$, $\lambda_{4}=21$.