Determine whether the indicated sets of complex vectors are...

Determine whether the indicated sets of complex vectors are linearly independent or dependent. (a) $\left(\begin{array}{l}\mathrm{i} \\ 1\end{array}\right),\left(\begin{array}{l}1 \\ \mathrm{i}\end{array}\right)$, (b) $\left(\begin{array}{c}1+\mathrm{i} \\ 1\end{array}\right),\left(\begin{array}{c}2 \\ 1-\mathrm{i}\end{array}\right)$, (c) $\left(\begin{array}{c}1+3 \mathrm{i} \\ 2-\mathrm{i}\end{array}\right),\left(\begin{array}{c}2-3 \mathrm{i} \\ 1-\mathrm{i}\end{array}\right)$, (d) $\left(\begin{array}{c}-2+\mathrm{i} \\ \mathrm{i}\end{array}\right),\left(\begin{array}{c}4-3 \mathrm{i} \\ 1\end{array}\right),\left(\begin{array}{c}2 \mathrm{i} \\ 1-5 \mathrm{i}\end{array}\right)$, (e) $\left(\begin{array}{c}1+2 \mathrm{i} \\ 2 \\ 0\end{array}\right),\left(\begin{array}{c}2 \\ 0 \\ 1-\mathrm{i}\end{array}\right)$, (f) $\left(\begin{array}{c}1 \\ 3 \mathrm{i} \\ 2-\mathrm{i}\end{array}\right),\left(\begin{array}{c}1+2 \mathrm{i} \\ -3 \\ 0\end{array}\right),\left(\begin{array}{c}1-\mathrm{i} \\ -\mathrm{i} \\ 1\end{array}\right)$, (g) $\left(\begin{array}{c}1+\mathrm{i} \\ 2-\mathrm{i} \\ 1\end{array}\right),\left(\begin{array}{c}1-\mathrm{i} \\ -3 \mathrm{i} \\ 1-2 \mathrm{i}\end{array}\right),\left(\begin{array}{c}-1+\mathrm{i} \\ 2+3 \mathrm{i} \\ 1+2 \mathrm{i}\end{array}\right)$.

Determine whether the indicated sets of complex vectors are linearly independent or dependent.
(a) $\left(\begin{array}{l}\mathrm{i} \\ 1\end{array}\right),\left(\begin{array}{l}1 \\ \mathrm{i}\end{array}\right)$,
(b) $\left(\begin{array}{c}1+\mathrm{i} \\ 1\end{array}\right),\left(\begin{array}{c}2 \\ 1-\mathrm{i}\end{array}\right)$,
(c) $\left(\begin{array}{c}1+3 \mathrm{i} \\ 2-\mathrm{i}\end{array}\right),\left(\begin{array}{c}2-3 \mathrm{i} \\ 1-\mathrm{i}\end{array}\right)$,
(d) $\left(\begin{array}{c}-2+\mathrm{i} \\ \mathrm{i}\end{array}\right),\left(\begin{array}{c}4-3 \mathrm{i} \\ 1\end{array}\right),\left(\begin{array}{c}2 \mathrm{i} \\ 1-5 \mathrm{i}\end{array}\right)$,
(e) $\left(\begin{array}{c}1+2 \mathrm{i} \\ 2 \\ 0\end{array}\right),\left(\begin{array}{c}2 \\ 0 \\ 1-\mathrm{i}\end{array}\right)$,
(f) $\left(\begin{array}{c}1 \\ 3 \mathrm{i} \\ 2-\mathrm{i}\end{array}\right),\left(\begin{array}{c}1+2 \mathrm{i} \\ -3 \\ 0\end{array}\right),\left(\begin{array}{c}1-\mathrm{i} \\ -\mathrm{i} \\ 1\end{array}\right)$,
(g) $\left(\begin{array}{c}1+\mathrm{i} \\ 2-\mathrm{i} \\ 1\end{array}\right),\left(\begin{array}{c}1-\mathrm{i} \\ -3 \mathrm{i} \\ 1-2 \mathrm{i}\end{array}\right),\left(\begin{array}{c}-1+\mathrm{i} \\ 2+3 \mathrm{i} \\ 1+2 \mathrm{i}\end{array}\right)$.

Key Concepts

Complex Vector Spaces

Complex vector spaces are vector spaces in which the field of scalars is the set of complex numbers. This allows vectors to have complex components, leading to operations such as complex addition and scalar multiplication. Understanding complex vectors is essential in many areas of mathematics and engineering, as they expand the usual real-number techniques to the complex domain.

Linear Independence

Linear independence is a fundamental concept in linear algebra. A set of vectors is linearly independent if the only solution to the equation c1·v1 + c2·v2 + ... + cn·vn = 0 is when all the coefficients c1, c2, ..., cn are zero. This concept helps in identifying whether the vectors form a basis for a space or have redundant elements.

Linear Dependence

Linear dependence occurs when there exists a nontrivial combination of vectors that equals the zero vector. That is, if at least one vector in the set can be expressed as a linear combination of the others, the set is linearly dependent. Recognizing linear dependence is crucial for understanding the minimal number of vectors needed to span a space.

Determinant Method

The determinant method is a tool used to verify linear independence when dealing with sets of vectors that form a square matrix (such as two vectors in a 2-dimensional space or three vectors in a 3-dimensional space). If the determinant of the matrix composed of these vectors is nonzero, the vectors are linearly independent; if it is zero, they are linearly dependent.

Systems of Linear Equations

To determine if a set of vectors is linearly independent or dependent, one can set up a system of linear equations based on a linear combination equating to the zero vector. Solving this system, often using techniques like row reduction or substitution, will reveal if there exists any nontrivial solution (indicating dependence) or only the trivial solution (indicating independence).

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