Linear Algebra Test - T 3

Rsquared Mathematics

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Time: 45:00

Topic: Linear Algebra

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The Quiz contains a total of 10 Questions.
Part B(Single Correct) contains 5 questions, each of 3 marks and -0.75 marks for incorrect answers.
Part C(MSQ) contains 5 questions, each with 4.75 marks and no negative marking. The maximum score for the Test is 38.75. Best of Luck.

Part - B

1. If \(\left\{e_1, e_2, \ldots, e_7\right\}\) denotes the standard basis of \(\mathbf{R}^{7 \times 1}(\mathbf{R})\) and consider a \(7 \times 7\) matrix \(A=\left[e_1, e_7, e_3, e_4, e_6, e_5, e_2\right]\)

(a) \(A^2=I\)
(b) \(A^3=I\)
(c) \(A^4=I\)
(d) \(A^5=I\)

Explanation: Here \(A\) is a permutation matrix corresponding to \(\sigma=(27)(56)\)

2. \(S=\left\{A: A=\left[a_{i j}\right]_{6 \times 6}\right.\) where \(a_{i j}=0\) or \(3 i, j\) and \(\sum_{i=1}^6 a_{i j}=3, \sum_{j=1}^6 a_{i j}=3\)\

(a) \(\mathrm{S}\) has cardinality \(3!\)
(b) \(\mathrm{S}\) has cardinality 9
(c) \(\mathrm{S}\) has cardinality \(3^6\)
(d) \(\mathrm{S}\) has cardinality \(6!\)

Explanation: \(A\) is a permutation matrix

3. Under which scaler multiplication \(\mathbf{R}^2=\{(x, y): x, y \in \mathbf{R}\}\) is a vector space over \(\mathbb{R}\)

(a) \(\alpha(x, y)=(|\alpha| x,|\alpha| y)\)
(b) \(\alpha(x, y)=(0, \alpha y)\)
(c) \(\alpha(x, y)=(\alpha^2 x, \alpha^2 y)\)
(d) None of the above

Explanation: Use properties of vector space

4. Let \(V=M_{n \times n}(\mathbf{R})\) be a vector space over the field of real numbers. Consider

$$ \begin{aligned} & W_1=\left\{A=\left[a_{i j}\right]_{n \times n} \mid \frac{a_{i j}}{j}-\frac{a_{j i}}{j}=0\right\} \\ & W_2=\left\{A=\left[a_{i j}\right]_{n \times n} \mid \frac{a_{i j}}{i}+\frac{a_{j i}}{i}=0\right\} \end{aligned} $$

Be subspace of \(\mathrm{V}\), then \(\operatorname{dim}(W_1+W_2)\) is

(a) \(\frac{n(n-1)}{2}\)
(b) \(\frac{n(n+1)}{2}\)
(c) \(n^2\)
(d) None of the above

Explanation: They are subspaces of symm. And anti. Symm. Matrices.

5. How many \(11 \times 11\) complex matrices (upto similarity) are there having the characteristic polynomial \((x-2)^5(x-3)^3(x-4)^3\)

(a) 63
(b) 45
(c) 25
(d) 11

Explanation: Multiplication of partition of powers

Part - C

6. Choose the correct statement

(a.) If \(S=\left\{x_2+a_2 x_1, x_3+a_3 x_1, \ldots, x_n+a_n x_1\right\}\) is linearly dependent than \(T=\left\{x_1, x_2, \ldots, x_n\right\}\) is also linearly dependent.
(b.) If \(S=\left\{x_2+a_2 x_1, x_3+a_3 x_1, \ldots, x_n+a_n x_1\right\}\) is linearly dependent than \(T=\left\{x_1, x_2, \ldots, x_n\right\}\) may not be linearly dependent.
(c.) The set \(\{f(x), g(x), h(x)\}\) of non-zero polynomials in \(\mathbf{R}[x]\) is linearly independent if \(\operatorname{deg} f(x), \operatorname{deg} g(x)\) and \(\operatorname{deg} h(x)\) are distinct.
(d.) The set \(\{f(x), g(x), h(x)\}\) of polynomials in \(\mathbf{R}[x]\) is linearly independent only if \(\operatorname{deg} f(x), \operatorname{deg} g(x)\) and \(\operatorname{deg} h(x)\) are distinct.

Explanation: For \(d\), polynomials can be LI having the same degree.

7. Let \(T: P_2(\mathbf{R}) \rightarrow P_3(\mathbf{R})\) be a linear transformation defined by \(T(f(x))=3 f^{\prime}(x)+\int_0^x 2 f(t) d t\) and rank space, \(R(T)\) is spanned by \(\left\{\alpha x, 3+\beta x^2, 6 x+\gamma x^3\right\}\) then the value of \(\alpha+\beta+\gamma\) can be

(a.) \(\frac{10}{3}\)
(b.) \(\frac{11}{3}\)
(c.) \(\frac{7}{3}\)
(d.) \(\frac{13}{3}\)

Explanation: Can be any non-zero values such that the set remains LI.

8. If \(T: M_{2 \times 2}(\mathbf{R}) \rightarrow \mathbf{R}^4\), be a linear transformation defined by \(T(A)=\left(\operatorname{trace} A, \operatorname{trace} A^t, \operatorname{trace} A B, \operatorname{trace} B A\right)\) where \(B=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]\), then

(a.) Rank \(T=3\)
(b.) Rank \(T\) = Nulity \(T\)
(c.) \(\operatorname{Trace}(T)=4\)
(d.) Trace (\(T\))=2

Explanation: Find the linear transformation and its matrix by using the information about the trace of \(A\), its transpose, and \(AB, BA\).

9. Let \(T: V_n(\mathbf{C}) \rightarrow V_n(\mathbf{C})\) be a linear transformation. Consider the following statements

P: If \(W\) is an \(1\)-dimensional invariant subspace under \(T\), then \(T\) must have an eigenvalue.

Q: If \(T\) is diagonalizable, then \(T\) must have an \(1\)-dimensional invariant subspace under \(T\)

Then which of the following is correct -

(a.) \(P\) is true, \(Q\) is false
(b.) \(P\) is not true but \(Q\) is true
(c.) \(P\) and \(Q\) both are true
(d.) \(P\) and \(Q\) both are false

Explanation: In \(P\), \(W=span(X)\) is \(T\)-invariant, that is there exists a non zero vector \(X\) such that T(span(\(X\)))=span(\(X\)), thus \(T(X)=\lambda X\).

10. Let \(V=C[0,1]\), Let \(T: \mathrm{V} \rightarrow \mathrm{V}\) be a linear operator given by \(T(f)=\int_0^1 \cos (\pi[x-t]) f(t) d t\)

(a.) \(\mathrm{T}\) has no eigenvalues
(b.) \(\mathrm{T}\) has exactly one eigenvalue
(c.) \(\mathrm{T}\) has infinitely many eigenvalues
(d.) \(\mathrm{T}\) has one eigenvalue with GM 1

Explanation: Try to find the Basis of Range Space.