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Question 4.7.2: Finding a Coordinate Matrix Relative to a Standard Basis The...

Finding a Coordinate Matrix

Relative to a Standard Basis

The coordinate matrix of x in R² relative to the (nonstandard) ordered basis B = \left\{v_1, v_2\right\} = \left\{(1, 0), (1, 2)\right\} is

[x]_B =\left [ \begin{matrix} 3 \\ 2 \end{matrix} \right ].

Find the coordinate matrix of x relative to the standard basis B´ = \left\{u_1, u_2\right\} = \left\{(1, 0), (0, 1)\right\} .

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Because [x]_B =\left [ \begin{matrix} 3 \\ 2 \end{matrix} \right ]. you can write x = 3v_1 + 2v_2 = 3(1, 0) + 2(1, 2) = (5, 4).
Moreover, because (5, 4) = 5(1, 0) + 4(0, 1), it follows that the coordinate matrix of x relative to B´ is

[x]_B =\left [ \begin{matrix} 5 \\ 4 \end{matrix} \right ].
Figure 4.19 compares these two coordinate representations.

4.19

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