Solve your IGNOU Doubts

Question:

Consider the linear operator ? ∶ ℂ³ → ℂ³ , defined by

$T(z_1,z_2,z_3)=(z_1-iz_2,iz_1-2z_2+iz_3,-iz_2+z_3).$

i) Compute ? ∗ and check whether ? is self-adjoint

ii) Check whether ? is unitary.

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Question:

Let ? be the vector space of all real valued functions that are twice differentiable in ℝ and $S=\{\cos x,\sin x,x\cos x,x\sin x\}.$

Check that ? is a linearly independent set over ℝ. (Hint: Consider the equation $a_0\cos x+a_1\sin x+a_2 x\cos x+a_3 x\sin x.$

$\left(\text{Put}x=0,\pi,\frac{\pi}{2},\frac{\pi}{4},\text{etc.and find}a_i.\right)$

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Question:

Find the minimal polynomial of the matrix $\begin{bmatrix}2&1&0&1\\-1&0&0&1\\-2&-2&-1&3\\0&0&0&1\end{bmatrix}$

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Question:

Solve the folowing set of simultaneous equations using Cramer’s rule:

$x+2y+z=3$

$2x-y+2z=1$

3? + ? + ? = 0

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Question:

Find the eigenvalues and eigenvectors of the matrix $B=\begin{bmatrix}1&1&0\\-1&3&0\\1&-1&1\end{bmatrix}$ Is the matrix diagonalisable? Justify your answer.

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Question:

Let ? ∶ ℝ³ → ℝ³ be a linear operator and suppose the matrix of the operator with respect to the ordered basis

$B=\left\{\begin{bmatrix}1\\0\\0\end{bmatrix},\begin{bmatrix}1\\1\\0\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

$\begin{bmatrix}1&0&-1\\0&1&1\\1&0&1\end{bmatrix}$ . Find the matrix of the linear transformation with respect to the basis $B'=\left\{\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}1\\0\\-1\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}$

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Question:

Check that $B=\{1,2x+1,(x-1)^2\}$ is a basis for ?², the vector space of polynomials with real coefficients of degree ≤ 2.

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Question:

Let ? be any non-empty set and let ? (?) be the set of all real valued functions on ℝ. Define addition on $(f+g)(x)=f(x)+g(x)$ and scalar multiplication by $(\alpha\cdot f)(x)=\alpha f(x).$ Check that $(V(S),+,\cdot)$ is a vector space.

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Question:

Find the signature of the quadratic form $x_1^2-2x_2^2+3x_3^2$

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Question:

Find adjoint of the linear operator ? ∶ ℂ² → ℂ² defined by $T(z_1,z_2)=(z_2,z_1+iz_2)$ respect to the standard inner product on ℂ 2 .

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Question:

Let ?[0, 1] be the inner product space of continous real valued functions on the interval [0, 1] with the inner produc

$\langle f,g\rangle=\int_0^1 f(t)g(t)\,dt$

Find the inner product of the functions $f(t)=2t,\quad g(t)=\frac{1}{t^2+5}.$

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Question:

Check whethe $\begin{bmatrix}1\\1\\1\\1\end{bmatrix}$ is an eigenvector for the matrix $\begin{bmatrix}1&0&0&1\\0&1&1&0\\0&0&1&1\\0&1&0&1\end{bmatrix}$ What is the corresponding eigenvalue?

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Question:

Verify Cayley-Hamilton theorem for the matrix $A=\begin{bmatrix}1&-1\\0&2\end{bmatrix}$

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Question:

Let ? be a 2 × 3 matrix, ? be a 3 × 4 matrix and ? be a 3 × 2 matrix and ? be a 3 × 4 matrix. Is ?? + ??? defined? Justify your answer.

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Question:

Find the matrix of the linear transformation $T:\mathbb{R}^2\rightarrow\mathbb{R}^2\text{defined by}T(x,y)=(2y,x-y)$ with respect to the ordered basis{(0, −1), (−1, 0)}.