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[QUE/VS-10008]

Node id: 3788page

In \(C^3\) with scalar product \((x,y)=x^\dagger y\) find the adjoint of operator \(A\) where \(A\) is defined to be
\begin{equation}
    A  \begin{pmatrix}x_1\\ x_2\\ x_3 \end{pmatrix}
     = \begin{pmatrix}x_1+ i x_3\\ x_2+i x_1 - ix_3 \\ x_1-i x_3\end{pmatrix}
\end{equation}

 kapoor's picture 22-04-25 08:04:42 n

[QUE/VS-10007]

Node id: 3787page

In \(C^3\) with scalar product \((x,y)=x^\dagger y\) find the adjoint of operator \(A\) where \(A\) is defined to be
\begin{equation}
    A  \begin{pmatrix}x_1\\ x_2\\ x_3 \end{pmatrix}
     = \begin{pmatrix}x_1+i x_3\\ x_1+x_2+x_3 \\ x_1-ix_3\end{pmatrix}
\end{equation}

 kapoor's picture 22-04-25 08:04:40 n

[QUE/VS-02005]

Node id: 3707page

PRO/VS-02005

Check if the vectors \((1,1,1,-1),(1,1,-1,1),(1,-1,1,1)\), in \(R^4\), are linearly independent. Do they form a basis in \(R^4\)? If your answer is 'YES' give a proof, if your answer is 'NO' give
an example of a vector which is not a linear combination of these vectors.

 kapoor's picture 22-04-25 08:04:01 n

[QUE/VS-02004]

Node id: 3706page

PRO/VS-02004

Do the vectors  \((1,1,1,-1),(1,1,-1,1),(1,-1,1,1),(-1,1,1,1)\) form a basis in  \(R^4\)? If your answer is 'YES' give a proof, if your answer is 'NO' give an example of a vector which is not a linear combination of these vectors.

 kapoor's picture 22-04-25 08:04:56 n

[QUE/VS-02001]

Node id: 3703page

PRO/VS-02001

Prove that in \(R^3\), three vectors \(\vec{A}, \vec{B},\vec{C}\) are linearly independent if and only if \(\vec{A}\cdot\vec{B}\times\vec{C}\ne 0\). How does this result generalize to vector space \(C^n\)?

 

 kapoor's picture 22-04-25 08:04:02 n

[QUE/VS-02002]

Node id: 3704page

PRO/VS-02002


Prove that the set of matrices
\begin{equation*}
  \begin{pmatrix}
     1 & 0 \\ 1& 1
  \end{pmatrix}
,~
  \begin{pmatrix}
     0& 1 \\ 1& 1
  \end{pmatrix}
,~
  \begin{pmatrix}
     1 & 1 \\ 0& 1
  \end{pmatrix}
,  ~  \begin{pmatrix}
     1 & 1 \\ 1& 0
  \end{pmatrix},
\end{equation*}
is a basis in the real vector space of all \(2\times 2\)  real matrices.

 kapoor's picture 22-04-25 08:04:56 n

[QUE/VS-02003]

Node id: 3705page

PRO/VS-02003

Show that the time period of a relativistic oscillator as function of amplitude $a$ is given by $ T\approx \frac{2\pi}{\omega}\left[ 1 + \frac{3\omega^2a^2}{16 c^2} + \cdots \right]  $  where the restoring force is $- kx$ and $k=m \omega2 $.

 kapoor's picture 22-04-25 08:04:17 n

[QUE/VS-01002]

Node id: 3699page

Consider the set of all vectors \(3\times3\) real matrices \(A\) for which

  1. \(\text{Tr}(A)=0\)
  2. \(\det A=0\)
  3. \(A_{11}>0\)
  4.  \(A_{11}, A_{22}, A_{33}\), all equal to zero
  5. \(A^{\text{T}}=A\)
  6. \(A^\text{T}=-A\)

Here \(A^\text{T}\) means the transpose of the matrix \(A\). In which of these cases do the set of all matrices \(A\)  form a vector space?

 kapoor's picture 22-04-25 08:04:46 n

[QUE/VS-01003]

Node id: 3700article

Consider the set of all polynomials \(p(t)= a_0 +a_1 t + a_2 t^2\) for
which

  1. \(p(0)=0\)
  2. \(2p(0)=p(1)\)
  3. \(p(t)=p(1-t)\)
  4. \(p(1) > 0\).

In which of these cases do the set of polynomials \(p(t)\) form a vector
space ?

 kapoor's picture 22-04-25 08:04:13 n

[QUE/VS-02007]

Node id: 3709page

PRO/VS-02007

Consider the set of all vectors \(\xi=(\xi_1,\xi_2,\xi_3)\) in \(C^3\) for which

  1. \(\xi_1\) is real
  2. \(\xi=0\)
  3. \(|\xi_1|> 0\)
  4. either \(\xi_1\) or \(\xi_2\) equal to zero
  5. \(\xi_1+\xi_2=0\)
  6. \(\xi_1+\xi_2=1\)


Give the dimensions of the vector spaces wherever appropriate and give a possible basis?

 kapoor's picture 22-04-25 08:04:49 n

[QUE/VS-02010]

Node id: 3712page

PRO/VS-02010

Consider the set of all polynomials \(p(t)= a_0 +a_1 t + a_2 t^2\) for which

  1. \(p(0)=0\)
  2. \(2p(0)=p(1)\)
  3. \(p(t)=p(1-t)\)
  4. \(p(1) > 0\).

Give the dimensions of the vector spaces wherever appropriate and give a possible basis?

 kapoor's picture 22-04-22 16:04:07 n

[QUE/VS-02009]

Node id: 3711page

PRO/VS-2009

Consider the set of all vectors \(3\times3\) real matrices \(A\) for which

  1. \(\text{Tr}(A)=0\)
  2. \(\det A=0\)
  3. \(A_{11}>0\)
  4. either \(A_{11}, A_{22}, A_{33}\), all equal to zero
  5. \(A^{\text{T}}=A\)
  6. \(A^\text{T}=-A\)


 Give the dimensions of the vector spaces wherever appropriate and give a possible basis?

 kapoor's picture 22-04-22 16:04:31 n

[QUE/VS-02011]

Node id: 3713page

PRO/VS-02011

Consider the linear span of the vectors (1,0,0), (1,1,0) and (1,-1,0).
What is the dimension of the vector space thus obtained?

 kapoor's picture 22-04-22 16:04:41 n

[QUE/VS-02012]

Node id: 3714page

PRO/VS-02012

Do the polynomials \(p_1(t)= 1-t, p_2(t)=t(1-t), p_3(t)=1-t^2\) give a basis for vector space  \(P_2(t)\) all polynomials ?

 kapoor's picture 22-04-22 16:04:44 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  \(\phi_1, \phi_2, \phi_3\).

 kapoor's picture 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 \(\Vsc_2\).

 kapoor's picture 22-04-22 15:04:35 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 \(P(t)/ V_1\).

 kapoor's picture 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 \[ \Rbb^3 = V_1 \oplus \Vsc_2 ?\]

 kapoor's picture 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.

 

 kapoor's picture 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*}
 kapoor's picture 22-04-22 15:04:34 n

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