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Title+Summary Name Date

### [QUE/VS-05002]

Node id: 3765page
1. Write as many criterion as you can which can be used to check if an operator is invertible.
2. Check if the operator $$A$$ on $$C^3$$ defined below is invertible. \begin{equation*} A\begin{pmatrix} x_1 \\ x_2 \\x_3 \end{pmatrix} = \begin{pmatrix} x_2 \\ x_3 \\x_1 \end{pmatrix} \end{equation*}
22-04-22 15:04:34 n

### [QUE/VS-05003]

Node id: 3766page

Show that the eigenvectors $$x_1, x_2, \ldots, x_n$$ of an operator $$A$$ corresponding to distinct eigenvalues $$\lambda_1, \lambda_2, \ldots\lambda_n$$ are linearly independent.

22-04-22 15:04:33 n

### [QUE/VS-05004]

Node id: 3767page

Show that real anti-symmetric matrices in odd dimensions are not invertible.

22-04-22 15:04:30 n

### [QUE/VS-06001]

Node id: 3768page

Find the matrix representation of the  operators $$A$$ on vector space of all polynomials of degree less than or equal to 3, where $Ap(t) = p(t+1),$ in the basis $$\{1,t,t^2, t^3\}$$.

22-04-22 15:04:47 n

### [QUE/VS-06002]

Node id: 3769page

Find the matrix representation of the  operators $$A$$ on the vector space of all  $$2\times 2$$ real matrices, where
$A \begin{pmatrix} a & b \\c & d\end{pmatrix} = \begin{pmatrix}1 & 1\\1& 1 \end{pmatrix} \begin{pmatrix} a & b \\c & d\end{pmatrix},$
in the basis
\begin{equation*}
e_1 = \begin{pmatrix}
1 & 0\\0& 0&0
e_2 = \begin{pmatrix}
0 & 1 \\ 0 & 0
e_3 = \begin{pmatrix}
0 & 0 \\ 1 & 0
e_4 = \begin{pmatrix}
0 & 0 \\
0 & 1
\end{pmatrix}.
\end{equation*}

22-04-22 15:04:02 n

### [QUE/VS-3002]

Node id: 3743page

Let $$1, t,t^2$$ be chosen as basis in $$\Pbb_2(t)$$ and let $$\phi_1, \phi_2, \phi_3$$ be the dual basis. Find the components of the linear functional  $\phi(p) = \int_{-1}^1 dt p(t)$ in the basis  $$\phi_1, \phi_2, \phi_3$$.

22-04-22 16:04:30 n

### [QUEVS-04001]

Node id: 3744page

Consider the vector space $$\Pbb_5(t)$$ whose elements $$p(t)$$ are polynomials in $$t$$ of degree less than or equal to 4.
$p(t) =\alpha_0 +\alpha_1t + \alpha_2 t^2 + \alpha_3 t^3 +\alpha_4t^4$  Consider the subspace
$$\Vsc_1$$ of $$\Pbb_5(t)$$ consisting of polynomials which are even functions  of  $$t$$. What is the dimension of $$\Vsc_1$$?  What is the vector space $$\Vsc_2$$ such that $$\Pbb_5(t) = \Vsc_1 \oplus \Vsc_2$$. Give a basis in $$\Vsc_2$$.

22-04-22 15:04:35 n

### [SHQ/QM-06002] --- Allowed outcomes of measurement

Node id: 1965page
22-04-17 21:04:25 n

### $$\S\S$$ 7.3 Q[1] $$\int_0^\infty\frac{1}{(x^2+p^2)^2} dx$$

Node id: 1566page
Full details are written out for this problem. This includes a proof that certain integrals vanish when Darboux theorem is used. In solutions to all other similar problems of this section, some of these details, being repetitive in nature, are suppressed. It is hoped that, if required, the reader will be able to supply the details by consulting this solution with full details.
20-02-08 16:02:37 n

### $$\S\S$$ 7.3 Q[2] $$\int_0^\infty \frac{x}{x^4+1} dx$$

Node id: 1835page
22-08-26 11:08:24 n

### "Trees versus Forest"

Node id: 79page
21-05-21 01:05:42

### $\S\S 7.11$ Q[1] $\int_0^\infty \frac{(1-\cos a x)}{x^2}dx, \quad a >0$

Node id: 1921page

$$\S\S\ 7.11$$ : : [1]  [2]  [3]  [4]  [5]  [6]  [7]  [8]  [9]  [10]  [11]  [12]  [13]  [14]  [15]

21-05-20 01:05:09 n

### 1. Current Density and Current Conservation*

Node id: 6056page
24-03-03 10:03:44 n

### 1.2 Why Postulates ? --- Lectures given at Hyd Univ -2024 Refresher Course

Node id: 6061page
24-03-01 04:03:15 n

### 1.4 Exercise-01 --- Hyd Univ-2024 Referesher Course

Node id: 6057page
24-03-01 04:03:35 n

### 1.5 Exercise-02 --- Hyd Univ-2024 Referesher Course

Node id: 6064page
24-03-01 04:03:17 n

### 13. Lecture on Mechanics: Specification of State

Node id: 187page
19-12-28 03:12:22

### 18.1 Continuous Energy Solutions

Node id: 1476page
19-12-28 03:12:36 n

### 18.2 Integral equation

Node id: 1477page
19-12-28 03:12:36 n

### 18.3 Born Approximation

Node id: 1478page
19-12-28 03:12:36 n

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