Question

Check whether each of the following subsets of R3 is linearly independent.
i) {(1,2,3), (-1,1,2), (2,1,1)}. ii) {(3,1,2); (-1,-1,-3), (-4,-3-0)}.

02 Feb 2018
Answer :
Word Count : 461
We are asked to check linear independence of two subsets of $\mathbb{R}^3$ manually. Let's go step by step. --- ### Recall: A set of vectors $\{v_1, v_2, v_3\}$ in $\mathbb{R}^3$ is linearly independent if the only solution to: $$ c_1 v_1 + c_2 v_2 + c_3 v_3 = 0 $$ is $c_1 = c_2 = c_3 = 0$. --- ## (i) Vectors: $(1,2,3), (-1,1,2), (2,1,1)$ Set up the equation: $$ c_1(1,2,3) + c_2(-1,1,2) + c_3(2,1,1) = (0,0,0) $$ This gives a system of 3 equations: 1. $c_1 - c_2 + 2c_3 = 0$ (from x-component) 2. $2c_1 + c_2 + c_3 = 0$ (from y-component) 3. $3c_1 + 2c_2 + c_3 = 0$ (from z-component) --- Step 1: Solve first two equations. From eq1:
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