[Math] How to find the vector $x$ determined by the given coordinate vector $[x]_B$ and basis $B$

linear algebra

Find the vector $x$ determined by the given coordinate vector $[x]_B$ and given basis $B$:
$$
B
= \left\{
\begin{bmatrix} 3 \\ -5 \end{bmatrix},
\begin{bmatrix} -4 \\ 6 \end{bmatrix}
\right\},
\quad
[x]_B
= \begin{bmatrix} 5 \\ 3 \end{bmatrix}.
$$

I saw a couple different ways to approach this problem from tutorials on YouTube, and each led to a different answer. This is what I got:

 3    -4     |     5
-5     6     |     3

3 * 5 + -4 * 3 = 3
-5 * 5 + 6 * 3 = -7

A =    3
      -7

v1 =   3
      -5

v2 =  -4
       6

3 * v1 + -7 * v2 = x

=   9   +   28
   -15     -42

=   37
    57

Is that the correct answer?

Best Answer

This is largely a matter understanding the definition and notation. Typically if you have a basis $B=\{ \mathbf v_1, \mathbf v_2 \}$ and are told that $[\mathbf x]_B = (x_1, x_2)$ for real numbers $x_1$, $x_2$, then $\mathbf x = x_1 \mathbf v_1 + x_2 \mathbf v_2$. In our particular case, $$\mathbf x=5v_1 + 3v_2 = 5(3, -5) + 3(-4, 6) = (15, -25) + (-12, 18) = (3, -7) = 3 \mathbf e_1 - 7 \mathbf e_2$$ where $\mathbf e_1 = (1, 0)$ and $\mathbf e_2 = (0, 1)$ are the standard basis vectors for $\mathbb R^2$.

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