Math Challenges

Submissions for Problem #7

Problem #7

Is the vector b a linear combination of the column vectors of matrix A?

\[ A=\begin{pmatrix}1 & -4 & 6\\ 0 & 4 & 8\\ -1 & 2 & -10\end{pmatrix}\mathbf{b}=\begin{pmatrix}3\\ -1\\ -3\end{pmatrix} \]

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zapwai

Solution: \( \begin{pmatrix}1 & -4 & 6 & 3\\ 0 & 4 & 8 & -1\\ -1 & 2 & -10 & -3\end{pmatrix}\Rightarrow\begin{pmatrix}1 & -4 & 6 & 3\\ 0 & 4 & 8 & -1\\ 0 & -2 & -4 & 0\end{pmatrix}\Rightarrow\begin{pmatrix}1 & -4 & 6 & 3\\ 0 & 4 & 8 & -1\\ 0 & 0 & 0 & -1\end{pmatrix} \)

Explanation:
Another way to phrase this question is to ask if there is a solution to the matrix equation Ax = b. Since the determinant here is zero, we have to actually row reduce to see whether the system is consistent or not. Here we have an inconsistent system, so it is not possible to express b as a multiple of the columns of A. No, b is not a linear combination of these column vectors.