Let e 1=pmatrix1\\0\\0pmatrix, e 2=pmatrix1\\1\\0pmatrix, e 3=pmatrix1\\1\\1pmatrix. For S=\ e 1, e 2, e 3\ in R3, which is correct?

$\sum_{i=1}^3|e_i\rangle\langle e_i|=I$

GATE 2020 PH – Question 37

Mathematical Physics · Linear vector spaces: basis, orthogonality and completeness · 2 marks · Multiple choice

Let $$|e_1\rangle=\begin{pmatrix}1\\0\\0\end{pmatrix},\quad |e_2\rangle=\begin{pmatrix}1\\1\\0\end{pmatrix},\quad |e_3\rangle=\begin{pmatrix}1\\1\\1\end{pmatrix}.$$ For $S=\{|e_1\rangle,|e_2\rangle,|e_3\rangle\}$ in $\mathbb R^3$, which is correct?

S is orthonormal.
S is linearly dependent.
S is a basis for $\mathbb R^3$.
$\sum_{i=1}^3|e_i\rangle\langle e_i|=I$

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Show answer and explanation

Correct answer: (C) S is a basis for $\mathbb R^3$.

Explanation

The matrix with these vectors as columns is upper triangular with diagonal entries 1, so its determinant is 1. Its columns are linearly independent, and three independent vectors span $\mathbb R^3$, making S a basis. They are not orthogonal, for example $e_1\cdot e_2=1$, so the orthonormal and identity-resolution claims fail. Select C.

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