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# Solving ${A}{x}={y}$ by Inspection

Sometimes it is possible to solve a system of linear equations by inspection. This requires some degree of familiarity with the column space of a matrix.

The expression ${A}{x}={y}$ denotes saying that ${y}$ is a linear combination of the columns of ${A}$ or that equivalently it is in the column space of ${A}$. The amounts by which columns of ${A}$ contribute to the linear combination are determined by the elements of ${x}$. Knowing this enables us to solve for ${x}$ by inspection.

If ${y}$ is in the column space of ${A}$ then there is at least one solution; otherwise, there is no solution.

The column space of ${A}$ is the vector space containing all possible linear combinations of the columns of ${A}$.

Here are some examples.

## Example 1

$\left(\begin{array}{cccc}3& 3& 1& 2\end{array}\right)$$\left(\begin{array}{c}{x}_{0}\\ {x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array}\right)$ $=$ $\left(\begin{array}{c}14\end{array}\right)$

Here is one possible way of writing ${y}$ as a linear combination of the columns ${c}_{i}$ of ${A}$:

$\left(\begin{array}{c}14\end{array}\right)$ $=$ ${2}{c}_{0}+{2}{c}_{1}+{0}{c}_{2}+{1}{c}_{3}$

$\left(\begin{array}{c}14\end{array}\right)$ $=$

$\left(\begin{array}{c}14\end{array}\right)$ $=$

The solution ${x}$ is obtained by writing the coefficients of the linear combination as a column vector. This is a possible solution; there may be more.

## Example 2

$\left(\begin{array}{cccc}3& 0& 0& 1\\ 2& 3& 1& 1\end{array}\right)$$\left(\begin{array}{c}{x}_{0}\\ {x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array}\right)$ $=$ $\left(\begin{array}{c}5\\ 8\end{array}\right)$

Here is one possible way of writing ${y}$ as a linear combination of the columns ${c}_{i}$ of ${A}$:

$\left(\begin{array}{c}5\\ 8\end{array}\right)$ $=$ ${1}{c}_{0}+{1}{c}_{1}+{1}{c}_{2}+{2}{c}_{3}$

$\left(\begin{array}{c}5\\ 8\end{array}\right)$ $=$

$\left(\begin{array}{c}5\\ 8\end{array}\right)$ $=$

The solution ${x}$ is obtained by writing the coefficients of the linear combination as a column vector. This is a possible solution; there may be more.

## Example 3

$\left(\begin{array}{cccc}1& 3& 2& 2\\ 0& 3& 1& 0\\ 1& 2& 3& 2\end{array}\right)$$\left(\begin{array}{c}{x}_{0}\\ {x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array}\right)$ $=$ $\left(\begin{array}{c}8\\ 2\\ 10\end{array}\right)$

Here is one possible way of writing ${y}$ as a linear combination of the columns ${c}_{i}$ of ${A}$:

$\left(\begin{array}{c}8\\ 2\\ 10\end{array}\right)$ $=$ ${2}{c}_{0}+{0}{c}_{1}+{2}{c}_{2}+{1}{c}_{3}$

$\left(\begin{array}{c}8\\ 2\\ 10\end{array}\right)$ $=$

$\left(\begin{array}{c}8\\ 2\\ 10\end{array}\right)$ $=$

The solution ${x}$ is obtained by writing the coefficients of the linear combination as a column vector. This is a possible solution; there may be more.

## Example 4

$\left(\begin{array}{cccc}2& 0& 3& 1\\ 1& 3& 0& 0\\ 3& 0& 3& 0\\ 0& 2& 3& 2\end{array}\right)$$\left(\begin{array}{c}{x}_{0}\\ {x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array}\right)$ $=$ $\left(\begin{array}{c}8\\ 2\\ 9\\ 5\end{array}\right)$

Here is one possible way of writing ${y}$ as a linear combination of the columns ${c}_{i}$ of ${A}$:

$\left(\begin{array}{c}8\\ 2\\ 9\\ 5\end{array}\right)$ $=$ ${2}{c}_{0}+{0}{c}_{1}+{1}{c}_{2}+{1}{c}_{3}$

$\left(\begin{array}{c}8\\ 2\\ 9\\ 5\end{array}\right)$ $=$

$\left(\begin{array}{c}8\\ 2\\ 9\\ 5\end{array}\right)$ $=$

The solution ${x}$ is obtained by writing the coefficients of the linear combination as a column vector. This is a possible solution; there may be more.

[This content was created with ML, available from the App Store.]