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### [QUE/VS-03001]

Node id: 3742page

In $$M^3(C)$$ consisting of all $$3\times3$$ complex matrices $$A= \begin{pmatrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{pmatrix}$$ which of the following are linear functionals?

1. $$\phi(A) = \text{Tr}(A)$$
2. $$\phi(A)= \text{Tr}(a) + \text{Tr}(A^T)$$
3. $$\phi(A) =\det (A)$$
4. $$\phi(A)= A_{11}+A_{21} + A_{31}$$
5. $$\phi(A) = A_{11}.$$
6. $$\phi(A)= A_{12}A_{32}$$
22-04-22 15:04:40 n

### [QUE/VS-04002]

Node id: 3745page

Consider the vector space $$P_5(t)$$ 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 $$V_1$$ of $$P_5(t)$$ consisting of polynomials  which are even functions of  $$t$$. What is the dimension of $$V_1$$?  What is the vector space $$V_2$$ such
that $$P_5(t) = V_1 \oplus V_2$$. What is dimension of quotient space $$P(t)/V_1$$? Give a basis for $$P(t)/ V_1$$.

22-04-22 15:04:51 n

### [QUE/VS-04003]

Node id: 3746page

Consider the vector spaces $$V_1$$ and $$V_2$$ obtained  by taking all possible real linear combinations of the vectors
$$(1.0,0),(1,1,0)$$ and of $$(0,1,0),(0,1,1)$$. Is it true that $\Rbb^3 = V_1 \oplus \Vsc_2 ?$

22-04-22 15:04:12 n

### [QUE/VS-05001]

Node id: 3763page

Find the basis in which the operator $$A$$ on $$C^3$$ defined by
\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*}
is diagonal.

22-04-22 15:04:38 n

### [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
\end{pmatrix};\quad
e_2 = \begin{pmatrix}
0 & 1 \\ 0 & 0
\end{pmatrix};\quad
e_3 = \begin{pmatrix}
0 & 0 \\ 1 & 0
\end{pmatrix};\quad
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

### [Solved/OP-02001] Normalization of Legendre Polynomials using Generating Function

Node id: 3659pdf_document
22-02-27 09:02:01 n

### $$\S 12.2$$ :: Potentials as Four Vectors

Node id: 2654pdf_document
20-02-08 16:02:10 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

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

Node id: 187page
19-12-28 03:12:22

### 16-Flu-Mech-1.17120

Node id: 5239pdf_document
22-02-09 21:02:32 n

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