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

1. Consider the quadratic form \(Q(v)=v^t A v\), where

$$A=\begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0\end{bmatrix}, v=(x, y, z, w)$$

Then

(a.) \(Q\) has rank 3
(b.) \(x y+z^2=Q(P v)\) for some invertible \(4 \times 4\) real matrix \(P\)
(c.) \(x y+y^2+z^2=Q(P v)\) for some invertible \(4 \times 4\) real matrix \(P\)
(d.) \(x^2+y^2-z w=Q(P v)\) for invertible \(4 \times 4\) real matrix \(P\)

2. Let \(u, v, w\) be vectors in an inner-product space \(V\), satisfying \(\|u\|=\|v\|=\|w\|=2\) and \(\langle u, v\rangle=0,\langle u, w\rangle=1,\langle v, w\rangle=-1\). Then which of the following are true?

(a.) \(\|w+v-u\|=2 \sqrt{2}\)
(b.) \(\left\{\frac{1}{2} u, \frac{1}{2} v\right\}\) forms an orthonormal basis of a two-dimensional subspace of \(V\)
(c.) \(w\) and \(4 u-w\) are orthogonal to each other.
(d.) \(u, v, w\) are necessarily linearly independent.

3. Let \(V\) denote the vector space of all polynomials over \(R\) of degree less than or equal to \(n\). Which of the following defines a norm on \(V\)?

(a.) \(\|p\|^2=|p(1)|^2+\ldots+|p(n+1)|^2\), \(p \in V\)
(b.) \(\|p\|=\sup _{t \in[0,1]}|p(t)|\), \(p \in V\)
(c.) \(\|p\|=\int_0^1|p(t)| d t\), \(p \in V\)
(d.) \(\|p\|=\sup _{t \in[0,1]}\left|p^{\prime}(t)\right|\), \(p \in V\)

4. Consider \(\mathbb{R}^3\) with the standard inner product. Let \(W\) be the subspace of \(\mathbb{R}^3\) spanned by \((1,0,-1)\). Which of the following is a basis for the orthogonal complement of \(W\)?

(a.) \(\{(1,0,1),(0,1,0)\}\)
(b.) \(\{(1,2,1),(0,1,1)\}\)
(c.) \(\{(2,1,2),(4,2,4)\}\)
(d.) \(\{(2,-1,2),(1,3,1),(-1,-1,-1)\}\)

5. Consider the quadratic forms \(q\) and \(p\) given by \(q(x, y, z, w)=x^2+y^2+z^2+b w^2\) and \(p(x, y, z, w)=x^2+y^2+c w z\). Which of the following statements are true?

(a.) \(p\) and \(q\) are equivalent over \(\mathbb{C}\) if \(b\) and \(c\) are non-zero complex numbers.
(b.) \(p\) and \(q\) are equivalent over \(\mathbb{R}\) if \(b\) and \(c\) are non-zero real numbers.
(c.) \(p\) and \(q\) are equivalent over \(\mathbb{R}\) if \(b\) and \(c\) are non-zero real numbers with \(b\) negative.
(d) \(p\) and \(q\) are NOT equivalent over \(\mathbb{R}\) if \(c=0\)

6. Let \(V\) be the inner product space consisting of linear polynomials, \(p:[0,1] \rightarrow \mathbf{R}\) (i.e. \(V\) consists of polynomials \(p\) of the form \(p(x)=a x+b, a, b \in \mathbb{R}\)), with the inner product defined by \(\langle p, q\rangle=\int_0^1 p(x) q(x) d x\) for \(p, q \in V\). An orthonormal basis of \(V\) is

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

7. Let \(a, b, c\) be positive real numbers such that \(b^2+c^2

(a.) All the eigenvalues of \(A\) are negative
(b.) All the eigenvalues of \(A\) are positive real
(c.) \(A\) can have a positive as well as a form negative eigenvalue
(d.) Eigenvalues of \(A\) can be non-real complex numbers.

8. Let \(A=\left(a_{i j}\right)\) be an \(n \times n\) complex matrix and let \(A^*\) denote the conjugate transpose of \(A\). Which of the following statements are necessarily true?

(a.) If \(A\) is invertible, then \(\operatorname{tr}\left(A^* A\right) \neq 0\), i.e., the trace of \(A^* A\) is non zero.
(b.) If \(\operatorname{tr}\left(A^* A\right) \neq 0\), then \(A\) is invertible.
(c.) If \(\left|\operatorname{tr}\left(A^* A\right)\right|
(d.) If \(\operatorname{tr}\left(A^* A\right)=0\), then \(A\) is the zero-matrix.

9. Suppose \(A\) is a \(3 \times 3\) symmetric matrix such that \([x, y, 1] A\begin{bmatrix}x \\ y \\ 1\end{bmatrix}=x y-1\). Let \(p\) be the number of positive eigenvalues of \(A\) and let \(q=\operatorname{rank}(A)-p\). Then

(a.) \(p=1\) Define
(b.) \(p=2\)
(c.) \(q=2\)
(d.) \(q=1\).

10. Let \(\xi\) be a primitive cube root of unity. Define \(A=\begin{bmatrix}\xi^{-1} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \xi\end{bmatrix}\). define \(v=\begin{pmatrix}v_1 \\ v_2 \\ v_3\end{pmatrix} \in \mathbb{R}^3\) define \(\mid |v_A||=\sqrt{\left|v A v^T\right|}\) where \(v^T\) is transpose of \(v\). If \(w=(1,1,1)\) then \(\mid w|_A\) equals

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