(a) Determine whether the vectors 𝐯1=( 1 i 0 ), 𝐯2=( 0 1+i 2 ), 𝐯3=

Step 1: **Check Linear Independence**u000ATo determine if the vectors $u005Cmathbf{v}_1, u005Cmathbf{v}_2, u005C

Question

(a) Determine whether the vectors $\mathbf{v}_1=\left(\begin{array}{l}1 \\ \mathrm{i} \\ 0\end{array}\right), \mathbf{v}_2=\left(\begin{array}{c}0 \\ 1+\mathrm{i} \\ 2\end{array}\right), \mathbf{v}_3=\left(\begin{array}{c}-1+\mathrm{i} \\ 1+\mathrm{i} \\ -1\end{array}\right)$, are linearly independent or linearly dependent. (b) Do they form a basis of $\mathbb{C}^3$ ? (c) Compute the Hermitian norm of each vector. (d) Compute the Hermitian dot products between all different pairs. Which vectors are orthogonal?

    (a) Determine whether the vectors $\mathbf{v}_1=\left(\begin{array}{l}1 \\ \mathrm{i} \\ 0\end{array}\right), \mathbf{v}_2=\left(\begin{array}{c}0 \\ 1+\mathrm{i} \\ 2\end{array}\right), \mathbf{v}_3=\left(\begin{array}{c}-1+\mathrm{i} \\ 1+\mathrm{i} \\ -1\end{array}\right)$, are linearly independent or linearly dependent. (b) Do they form a basis of $\mathbb{C}^3$ ? (c) Compute the Hermitian norm of each vector. (d) Compute the Hermitian dot products between all different pairs. Which vectors are orthogonal?

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 3, Problem 28

Instant Answer

Step 1

We can do this by forming a matrix with $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ as columns and row reducing it. Form the matrix: \[ A = \begin{bmatrix} 1 & 0 & -1+\mathrm{i} \\ \mathrm{i} & 1+\mathrm{i} & 1+\mathrm{i} \\ 0 & 2 & -1 \end{bmatrix} \] Step 2: Show more…

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(a) Determine whether the vectors $\mathbf{v}_1=\left(\begin{array}{l}1 \\ \mathrm{i} \\ 0\end{array}\right), \mathbf{v}_2=\left(\begin{array}{c}0 \\ 1+\mathrm{i} \\ 2\end{array}\right), \mathbf{v}_3=\left(\begin{array}{c}-1+\mathrm{i} \\ 1+\mathrm{i} \\ -1\end{array}\right)$, are linearly independent or linearly dependent. (b) Do they form a basis of $\mathbb{C}^3$ ? (c) Compute the Hermitian norm of each vector. (d) Compute the Hermitian dot products between all different pairs. Which vectors are orthogonal?

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