Question

Suppose A is a \(3 \times 3\) matrix and b is a vector in \({\mathbb{R}^3}\) with the property that \(A{\bf{x}} = {\bf{b}}\) has a unique solution. Explain why the columns of A must span \({\mathbb{R}^3}\).

Short answer

The columns of A span \({\mathbb{R}^3}\), and the general echelon form of the given vector is \(A = \left[ {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\0&0&1\end{array}} \right]\).

Step-by-step solution

Writing the definition of \(A{\bf{x}}\)

The column of matrix \(A\) is represented as \(\left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{ \cdot \cdot \cdot }&{{a_n}}\end{array}} \right]\), and vector x is represented as \(\left[ {\begin{array}{*{20}{c}}{{x_1}}\\ \vdots \\{{x_n}}\end{array}} \right]\).

According to the definition, the weights in a linear combination of matrix A columns are represented by the entries in vector x.

The matrix equation as a vector equation can be written as shown below:

\(\begin{array}{c}A{\bf{x}} = \left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{ \cdot \cdot \cdot }&{{a_n}}\end{array}} \right)\left[ {\begin{array}{*{20}{c}}{{x_1}}\\ \vdots \\{{x_n}}\end{array}} \right]\\b = {x_1}{a_1} + {x_2}{a_2} + \cdots + {x_n}{a_n}\end{array}\)

The number of columns in matrix \(A\) should be equal to the number of entries in vector x so that \(A{\bf{x}}\) can be defined.

Identifying the conditions for a unique solution

There should be no free variable in the associated system of equations when the equation \(A{\bf{x}} = {\bf{b}}\) has a unique solution. Each column of A is a pivot column if every variable is a basis.

Writing the conditions for echelon and reduced echelon forms

The matrix is in echelon form if it satisfies the following conditions:

• Non-zero rows should be positioned above zero rows.
• Each row's leading entry should bein the column to the right of the row above its leading item.
• In each column, all items below the leading entry must be zero.

For reduced echelon form, the matrix must follow some additional conditions:

• Each column's components below the leading entry must be zero.
• Each column's leading 1 must be the sole non-zero item.

Writing the reduced echelon form

Consider a \(3 \times 3\) matrix in the reduced echelon form as shown below:

\(A = \left[ {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\0&0&1\end{array}} \right]\)

Here, each column's components below the leading entry are zero, and each column's leading 1 is the sole non-zero item.

Matrix Ahas enough pivot positions in each row. It means that the columns of A span \({\mathbb{R}^3}\).

Thus, the general echelon form of vector is \(A = \left[ {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\0&0&1\end{array}} \right]\).