Find the vector $x$ determined by the given coordinate vector $_B$ and given basis $B$: $$ B = left eginbmatrix 3 \ -5 endbmatrix, eginbmatrix -4 \ 6 endbmatrix ight, quad _B = eginbmatrix 5 \ 3 endbmatrix.$$

I observed a couple different means to method this problem from tutorials on YouTube, and also each resulted in a different answer. This is what I got:

3 -4 | 5-5 6 | 33 * 5 + -4 * 3 = 3-5 * 5 + 6 * 3 = -7A = 3 -7v1 = 3 -5v2 = -4 63 * v1 + -7 * v2 = x= 9 + 28 -15 -42= 37 57Is that the correct answer?




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This is greatly a matter knowledge the definition and notation. Usually if you have actually a basis $B= astecraftedmcd.combf v_1, astecraftedmcd.combf v_2 $ and are told that $< astecraftedmcd.combf x>_B = (x_1, x_2)$ for genuine numbers $x_1$, $x_2$, then $ astecraftedmcd.combf x = x_1 astecraftedmcd.combf v_1 + x_2 astecraftedmcd.combf v_2$. In our particular instance, $$ astecraftedmcd.combf x=5v_1 + 3v_2 = 5(3, -5) + 3(-4, 6) = (15, -25) + (-12, 18) = (3, -7) = 3 astecraftedmcd.combf e_1 - 7 astecraftedmcd.combf e_2$$ wright here $ astecraftedmcd.combf e_1 = (1, 0)$ and $ astecraftedmcd.combf e_2 = (0, 1)$ are the traditional basis vectors for $ astecraftedmcd.combb R^2$.


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