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Solve the system of equations α(1 1 1) + β(3 2 1) + γ(1 1 0) + δ(1 0 0) = (a b c) for arbitrary a, b, and c. If there is always a solution, then the vectors span R3; if there is a choice of a, b, c for which the system is inconsistent, then the vectors do not span R3.
To determine if a set of vectors is linearly independent, follow these steps: Consider a set of vectors, \mathbf {\vec {v_1}},\mathbf {\vec {v_2}},\ldots,\mathbf {\vec {v_n}} v1.
In order to show a set is linearly independent, you start with the equation c₁x⃑₁ + c₂x⃑₂ + ... + cₙx⃑ₙ = 0⃑ (where the x vectors are all the vectors in your set) and show that the only solution is that c₁ = c₂ = ... = cₙ = 0. If you can show this, the set is linearly independent.
Sep 17, 2022 · Determine the span of a set of vectors, and determine if a vector is contained in a specified span. Determine if a set of vectors is linearly independent. Understand the concepts of subspace, basis, and dimension. Find the row space, column space, and null space of a matrix.
Linearly independent vectors with examples. A set of vectors is linearly independent when none of the vectors can be written as a linear combination of the other vectors. This applies to vectors in \ (\mathbb {R}^n\) for any \ (n\) or vector spaces like the polynomial spaces.
Sep 26, 2012 · If you solve this system (say, by Gaussian elimination) you will find that it has non zero solution (for example $r_1=1,r_2=-1,r_3=1,r_4=-1$) so $u_1-u_2+u_3-u_4=0$ and your vectors are linearly depented.
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