Linear Algebra Test - T 1

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. Let \(V\) be a 3-dimensional vector space over the field \(\frac{\mathbb{Z}}{3 \mathbb{Z}}\) of 3 elements. The number of distinct 1-dimensional subspaces of \(V\) is

(a) 13
(b) 26
(c) 9
(d) 15

Exp. No. of Subspace for any \(V\) of dimension \(d=\frac{\prod_{d=0}^{d=r-1} p^{n k}-p^{d k}}{\prod_{d=0}^{d=r-1} p^{r k}-p^{d k}}\)

2. There exists an Idempotent matrix \(A_{5 \times 5}\) such that

(a) \(\operatorname{Trace}(\mathrm{A})=3\) and \(\operatorname{Rank}(\mathrm{A})=0\)
(b) \(\operatorname{Trace}(\mathrm{A})=7\)
(c) \(\operatorname{Rank}(\mathrm{A})=4\) and \(\operatorname{Trace}(\mathrm{A})=4\)
(d) \(\operatorname{Rank}(\mathrm{A})=2\) and \(\operatorname{Trace}(\mathrm{A})\) is negative

Exp. For idempotent matrices \(-\operatorname{Rank}(\mathrm{A})=\operatorname{Trace}(\mathrm{A})\)

3. Find the Value of \(b\) for which the Quadratic form is Positive definite

\( Q(x, y, z)=2 x^{2}+(b+4) y^{2}+(b+1) z^{2}+2 b x y+2(b-4) y z \)

(a) \(1 < b < 2\)
(b) \(1 < b < 4\)
(c) \(-1 < b < 4\)
(d) \(-1 < b < 2\)

Exp. Here, \(\mathrm{A}=\left(\begin{array}{ccc}2 & b & 0 \\ b & b+4 & b-4 \\ 0 & b-4 & b+1\end{array}\right)\), select \(b=0\) and \(3\) and discard \(a, c, d\)

4. Suppose that \(A\) is a square matrix such that \(B^{T} A B\) is a diagonal matrix for an orthogonal matrix \(B\), Then

(a) \(A\) is necessarily symmetric matrix
(b) \(A\) is skew symmetric matrix
(c) \(A\) is Hermitian matrix
(d) \(A\) is skew Hermitian matrix

Exp. \(B^{T} A B\) is a diagonal matrix for an orthogonal matrix \(B\), Then \(A\) must be symmetric.

5. Which of the following matrix representation is of a linear operator

\( T: \mathbf{R}^{3} \rightarrow \mathbf{R}^{3} \text{ defined as } T(x, y, z)=(2 x, y+z, x+y+2 z) \text{ w.r.t. basis }\) \( \{(1,2,1),(1,1,1),(1,1,0)\} \)

(a) \(\left[\begin{array}{ccc}1 & -1 & 0 \\ 4 & 3 & 2 \\ -3 & 0 & 5\end{array}\right]\)
(b) \(\left[\begin{array}{ccc}1 & 0 & -1 \\ 4 & 4 & 3 \\ -3 & -2 & 0\end{array}\right]\)
(c) \(\left[\begin{array}{ccc}-1 & 0 & 1 \\ -4 & 4 & -3 \\ 3 & 2 & 0\end{array}\right]\)
(d) \(\left[\begin{array}{ccc}1 & 4 & -3 \\ 0 & 4 & 2 \\ -1 & 3 & 2\end{array}\right]\)

Part - C

6. Let \(A\) be a complex matrix \((5 \times 5)\) such that \(A^{3}=A\) then

(a.) \(\exists\) a vector \(x \neq 0\) such that \(x^{T} A x>0\)
(b.) \(\exists\) a vector \(x \neq 0\) such that \(x^{T} A x \leq 0\)
(c.) \(\exists\) a vector \(x \neq 0\) such that \(x^{T} A x=0\)
(d.) \(A\) is Diagonalizable over \(\mathbb{C}\)

Exp. \(-A^{3}=A \Rightarrow x^{3}-x=0\)
Possible minimal polynomials are \(x, x-1, x+1,(x^{2}-1), x(x^{2}-1)\), Hence \(A\) can be chosen as 0, Identity or \(-I\) and a, b, c can be discarded.

7. Let \(V=\{p(x) \mid p(x)\) is a polynomial over \(\mathbb{R}\}\) be a vector space over \(\mathbb{R}\) under usual addition & scalar multiplication of polynomials, then which of the following is/are not a sub-space of \(V\) -

(a.) \(W=\{p(x) \in V \mid \operatorname{deg} p(x)=4\}\)
(b.) \(W=\{p(x) \in V \mid \operatorname{deg} p(x) \geq 3\}\)
(c.) \(W=\left\{p(x) \in V \mid \operatorname{deg} p(x) \leq 4 p^{\prime}(0)=1\right\}\)
(d.) \(W=\{p(x) \in V \mid \operatorname{deg} p(x) \leq 3\}\)

Explanation: \(0 \notin W\) for a, b, and c.

8. If \(S=\{x, \sin x, \cos x,\{x\},[x]\} \subseteq V\), where \(V\) is a vector space of all functions on \(\mathbb{R}\) over \(\mathbb{R}\) and all other symbols have their usual meaning, then choose the correct -

(a.) \(S\) is \(\mathrm{LI}\) set
(b.) \(S\) is LD set
(c.) There exists a subset of \(\mathbb{R}\) on which \(S\) is LD
(d.) \(S\) is LI over every subset of \(\mathbb{R}\)

Exp. \(\{x\}=x-[x]\)

9. Let \(A=\left[a_{i j}\right]_{2020 \times 2020} ; a_{i j} \in \mathbb{R}\) such that \(A^{1010}=0\) and \(A^{r} \neq 0\) if \(r<1010\) and \(\mathrm{k}\) denote the rank of matrix \(A\), Then which of the following is/are true-

(a.) \(k=1009\)
(b.) \(k \geq 1010\)
(c.) \(1009 \leq k \leq 2018\)
(d.) \(k \neq 2019\)

Exp. Jordan form for nilpotent matrices.

10. Let \(A x=b\) be a non-homogeneous system of equations. If \(\mathrm{S}\) is the solution set of \(A x=b\) and \(S_{h}\) is the solution set of the associated homogeneous system \(A x=0\) then

(a.) if \(x \in S\) and \(y \in S_{h}\) and \(a \in \mathbf{R}\) implies \(x+a y \in S\)
(b.) if \(x, y \in S\) then \(x-y \in S_{h}\).
(c.) if \(x \in S\), then \(S=x+S_{h}=\left\{x+y: y \in S_{h}\right\}\)
(d.) All of the above.

Exp.
For a, if \(x \in S\) and \(y \in S_{h}\) and \(a \in \mathbf{R}\) implies \(\mathrm{A} x+A(a y)=b+a A y=b+0=b\)
For \(\mathrm{b}\), if \(x, y \in S\) then \(\mathrm{A} x-A y=b-b=0 \Rightarrow x-y \in S_{h}\)