Linear Algebra MCQs and MSQs with Solutions for CSIR NET, IIT JAM : Bilinear and Quadratic Forms - II

11. Consider the quadratic form \(q(x, y, z)=4 x^2+y^2-z^2+4 x y\) over \(\mathbb{R}\). Which of the following statements about range of values taken by \(q\) as \(x, y, z\) vary over \(\mathbb{R}\), are true?

(a.) Range contains \([1, \infty)\)
(b.) Range is contained in \([0, \infty)\)
(c.) Range \(=\mathbb{R}\)
(d.) Range is contained in \([-N, \infty)\) for some large natural number \(N\) depending on \(q\)

12. Let \(J\) be the \(3 \times 3\) matrix all of whose entries are 1. Then:

(a.) 0 and 3 are the only eigenvalues of \(J\)
(b.) \(J\) is positive semi-definite, i.e., matrix \(\langle J x, x\rangle \geq 0\) for all \(x \in \mathbb{R}^3\)
(c.) \(J\) is diagonalizable
(d.) \(J\) is positive definite, i.e., \(\langle J x, x\rangle>0\) for all \(x \in \mathbb{R}^3\) with \(x \neq 0\)

13. Let \(\zeta\) be a primitive fifth root of unity.

$$A=\begin{bmatrix}\zeta^{-2} & 0 & 0 & 0 & 0 \\ 0 & \zeta^{-1} & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & \zeta & 0 \\ 0 & 0 & 0 & 0 & \zeta^2\end{bmatrix}$$. If \(w=(1,-1,1,1,-1)\), then \(|w|_A\) equals

(a.) 0
(b.) 1
(c.) -1
(d.) 2

14. Let \(a_{i j}=a_i a_j, 1 \leq i, j \leq b\), where \(a_1, \ldots, a_n\) are real numbers. Let \(A=\left(\left(a_{i j}\right)\right)\) be the \(n \times n\) matrix \(\left(\left(a_{i j}\right)\right)\). Then

(a.) It is possible to choose \(a_1, \ldots ., a_n\) so as to make the matrix \(A\) non singular
(b.) The matrix \(A\) is positive definite if \(\left(a_1, \ldots, a_n\right)\) is a non zero vector
(c.) The matrix \(A\) is positive semi definite for all \(\left(a_1, \ldots, a_n\right)\)
(d.) For all \(\left(a_1, \ldots, a_n\right)\), zero is an eigenvalue of \(A\).

15. Suppose \(A, B\) are \(n \times n\) positive definite matrices and \(I\) be the \(n \times n\) identity matrix. Then which of the following are positive definite.

(a.) \(A+B\)
(b.) \(A B A^*\)
(c.) \(A^2+I\)
(d.) \(A B\)

16. Let \(T\) be a linear transformation on the real vector space \(\mathbf{R}^n\) over \(\mathbf{R}\) such that \(T^2=\lambda T\) for same \(\lambda \in \mathbf{R}\). Then

(a.) \(\|T x\|=|\lambda|\|x\|\) for all \(x \in \mathbf{R}^n\)
(b.) If \(\|T x\|=\|x\|\) for some non zero vector \(x \in \mathbf{R}^n\), then \(\lambda= \pm 1\)
(c.) \(T=\lambda I\) where \(\mathrm{I}\) is the identity transformation on \(\mathbf{R}^n\)
(d.) If \(\|T x\|>\|x\|\) for a non zero vector \(x \in \mathbf{R}^n\), then \(\mathrm{T}\) is necessarily singular.

17. The application of Gram-Schmidt process of orthonormalization to \(u_1=(1,1,0)\), \(u_2=(1,0,0)\), \(u_3=(1,1,1)\) yields

(a.) \(\frac{1}{\sqrt{2}}(1,1,0),(1,0,0),(0,0,1)\)
(b.) \(\frac{1}{\sqrt{2}}(1,1,0), \frac{1}{\sqrt{2}}(1,-1,0), \frac{1}{\sqrt{2}}(1,1,1)\)
(c.) \((0,1,0),(1,0,0),(0,0,1)\)
(d.) \(\frac{1}{\sqrt{2}}(1,1,0), \frac{1}{\sqrt{2}}(1,-1,0),(0,0,1)\)

18. Consider the basis \(\left\{u_1, u_2, u_3\right\}\) of \(\mathbf{R}^3\); where \(u_1=(1,0,0)\), \(u_2=(1,1,0)\), \(u_3=(1,1,1)\). Let \(\left\{f_1, f_2, f_3\right\}\) be the dual basis of \(\left\{u_1, u_2, w_3\right\}\) and \(f\) be a linear functional defined by \(f(a, b, c)=a+b+c\), \((a, b, c) \in \mathbf{R}^3\). If \(f=\alpha_1 f_1+\alpha_2 f_2+\alpha_3 f_3\). then \((\alpha_1, \alpha_2, \alpha_3)\) is

(a.) (1,2,3)
(b.) (1,3,2)
(c.) (2,3,1)
(d) (3,2,1)

19. Let \(\mathrm{M}=\begin{bmatrix}1 & 1 & 2 \\ 0 & 1 & 1 \\ 0 & 1 & 1\end{bmatrix}\) and \(V=\left\{M x^2: x \in R^3\right\}\). Then an orthonormal basis for \(V\) is

(a.) \(\left\{(1,0,0)^t,\left(0, \frac{2}{\sqrt{5}}, \frac{1}{\sqrt{5}}\right)^t \cdot\left(\frac{2}{\sqrt{6}}, \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}}\right)^t\right\}\)
(b.) \(\left\{(1,0,0)^t,\left(0, \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)^t\right\}\)
(c.) \(\left\{(1,0,0)^t,\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)^t\right.\).
(d) \(\left\{(1,0,0)^t,(0,0,1)^t\right\}\)

20. Consider \(\mathbf{R}^3\) with the standard inner product. Let \(S=\{(1,1,1),(2,-1,2),(1,-2,1)\}\). For a subset \(W\) of \(R^3\), let \(L(W)\) denote the linear span of \(W\) in \(R^3\). Then an arthonormal set \(\mathrm{T}\) with \(L(S)=L(T)\) is

(a.) \(\left\{\frac{1}{\sqrt{3}}(1,1,1), \frac{1}{\sqrt{6}}(1,-2,1)\right\} \)
(b.) \(\{(1,0,0),(0,1,0),(0,0,1)\}\)
(c.) \(\left\{\frac{1}{\sqrt{3}}(1,1,1), \frac{1}{\sqrt{2}}(1,-1,0)\right\}\)
(d.) \(\left\{\frac{1}{\sqrt{3}}(1,1,1), \frac{1}{\sqrt{2}}(0,1,-1)\right\}\)