Inner Products and Quadratic Forms - Linear Algebra - Mathematics Previous Year Questions

1

2008 · Mathematics · Linear Algebra · Inner Products and Quadratic Forms

Mathematics (MA) 2008

Let \(N = \begin{pmatrix} 3/5 & -4/5 & 0 \\ 4/5 & 3/5 & 0 \\ 0 & 0 & 1 \end{pmatrix}\). Then \(N\) is

A

non-invertible

B

skew-symmetric

C

symmetric

D

orthogonal

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2

2009 · Mathematics · Linear Algebra · Inner Products and Quadratic Forms

Mathematics (MA) 2009

Let \(V\) be the column space of the matrix \(A = \begin{pmatrix} 1 & -1 \\ 1 & 2 \\ 1 & -1 \end{pmatrix}\). Then the orthogonal projection of \(\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\) on \(V\) is

A

\(\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\)

B

\(\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}\)

C

\(\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}\)

D

\(\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}\)

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3

2010 · Mathematics · Linear Algebra · Inner Products and Quadratic Forms

Mathematics (MA) 2010

Let \( T : \mathbb{R}^3 \to \mathbb{R}^3 \) be a linear transformation defined by \( T(x, y, z) = (x + y, y + z, z - x) \). Then, an orthonormal basis for the range of \( T \) is

A

\(\left( \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0 \right), \left( \frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right) \)

B

\(\left( \frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}}, 0 \right), \left( \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}}, -\frac{2}{\sqrt{6}} \right) \)

C

\(\left( \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0 \right), \left( \frac{1}{\sqrt{6}}, -\frac{1}{\sqrt{6}}, -\frac{2}{\sqrt{6}} \right) \)

D

\(\left( \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0 \right), \left( \frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right) \)

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4

2011 · Mathematics · Linear Algebra · Inner Products and Quadratic Forms

Mathematics (MA) 2011

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

\(\frac{1}{\sqrt{2}} (1,1,0), \frac{1}{\sqrt{2}} (1,-1,0), (0,0,1)\)

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5

2011 · Mathematics · Linear Algebra · Inner Products and Quadratic Forms

Mathematics (MA) 2011

Let \( T : \mathbb{C}^3 \to \mathbb{C}^3 \) be defined by \( T \begin{pmatrix} z_1 \\ z_2 \\ z_3 \end{pmatrix} = \begin{pmatrix} z_1 - i z_2 \\ i z_1 + z_3 \\ z_1 + z_2 + i z_3 \end{pmatrix}. \) Then, the adjoint \( T^* \) of \( T \) is given by \( T^* \begin{pmatrix} z_1 \\ z_2 \\ z_3 \end{pmatrix} = \)

A

\(\begin{pmatrix} z_1 + i z_2 \\ -i z_1 + z_2 \\ z_1 + z_2 - i z_3 \end{pmatrix}\)

B

\(\begin{pmatrix} z_1 - i z_2 + z_3 \\ -i z_1 + z_2 + z_3 \\ i z_3 \end{pmatrix}\)

C

\(\begin{pmatrix} z_1 - i z_2 + z_3 \\ i z_1 + z_2 + z_3 \\ -i z_3 \end{pmatrix}\)

D

\(\begin{pmatrix} i z_1 + z_2 \\ z_1 - i z_2 \\ z_1 - z_2 - i z_3 \end{pmatrix}\)

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6

2012 · Mathematics · Linear Algebra · Inner Products and Quadratic Forms

Mathematics (MA) 2012

For the matrix
\( M = \begin{pmatrix} 2 & 3+2i & -4 \\ 3-2i & 5 & 6i \\ -4 & -6i & 3 \end{pmatrix} \),
which of the following statements are correct?
P : \( M \) is skew-Hermitian and \( iM \) is Hermitian
Q : \( M \) is Hermitian and \( iM \) is skew-Hermitian
R : eigenvalues of \( M \) are real
S : eigenvalues of \( iM \) are real

A

P and R only

B

Q and R only

C

P and S only

D

Q and S only

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7

2013 · Mathematics · Linear Algebra · Inner Products and Quadratic Forms

Mathematics (MA) 2013

Let \(B\) be a real symmetric positive-definite \(n \times n\) matrix. Consider the inner product on \(\mathbb{R}^n\) defined by \(\langle x, y \rangle = y^t Bx\). Let \(A\) be an \(n \times n\) real matrix and let \(T: \mathbb{R}^n \to \mathbb{R}^n\) be the linear operator defined by \(T(X) = AX\) for all \(X \in \mathbb{R}^n\). If \(S\) is the adjoint of \(T\), then \(S(X) = CX\) for all \(X \in \mathbb{R}^n\), where \(C\) is the matrix

A

\(B^{-1} A^t B\)

B

\(B A^t B^{-1}\)

C

\(B^{-1} A B\)

D

\(A^t\)

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8

2014 · Mathematics · Linear Algebra · Inner Products and Quadratic Forms

Mathematics (MA) 2014

Let V be a real inner product space of dimension 10. Let x, y ∈ V be non-zero vectors such that ⟨x, y⟩ = 0. Then the dimension of {x}⊥ ∩ {y}⊥ is __________

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9

2016 · Mathematics · Linear Algebra · Inner Products and Quadratic Forms

Mathematics (MA) 2016

Consider the following statements P and Q:
(P): If M is an n × n complex matrix, then ℛ(M) = (𝒩(M^*))^⊥.
(Q): There exists a unitary matrix with an eigenvalue λ such that |λ| < 1.
Which of the above statements hold TRUE?

A

both P and Q

B

only P

C

only Q

D

Neither P nor Q

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10

2018 · Mathematics · Linear Algebra · Inner Products and Quadratic Forms

Mathematics (MA) 2018

Consider \(\mathbb{R}^3\) with the usual inner product. If \(d\) is the distance from \((1,1,1)\) to the subspace span\(\{(1,1,0), (0,1,1)\}\) of \(\mathbb{R}^3\), then \(3d^2 = ________.\)

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11

2019 · Mathematics · Linear Algebra · Inner Products and Quadratic Forms

Mathematics (MA) 2019

Consider the inner product space \(P_2\) of all polynomials of degree at most 2 over the field of real numbers with the inner product \(\langle f, g \rangle = \int_0^1 f(t) g(t) dt\) for \(f, g \in P_2\).
Let \(\{f_0, f_1, f_2\}\) be an orthogonal set in \(P_2\), where \(f_0 = 1, f_1 = t + c_1, f_2 = t^2 + c_2 f_1 + c_3\) and \(c_1, c_2, c_3\) are real constants. Then the value of \(2c_1 + c_2 + 3c_3\) is equal to ______.

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12

2020 · Mathematics · Linear Algebra · Inner Products and Quadratic Forms

Mathematics (MA) 2020

Let \( M = \begin{bmatrix} \alpha & 3 & 0 \\ \beta & 3 & 1 \\ 0 & 1 & 2 \end{bmatrix} \). Consider the following statements:
I: There exists a lower triangular matrix \(L\) such that \(M = LL^t\), where \(L^t\) denotes transpose of \(L\).
II: Gauss-Seidel method for \(Mx = b\) (\(b \in \mathbb{R}^3\)) converges for any initial choice \(x_0 \in \mathbb{R}^3\).
Then

A

I is not true when \(\alpha > \frac{9}{2}\), \(\beta = 3\)

B

II is not true when \(\alpha \geq \frac{9}{2}\), \(\beta = -1\)

C

II is not true when \(\alpha = 4\), \(\beta = \frac{3}{2}\)

D

I is true when \(\alpha = 5\), \(\beta = 3\)

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13

2021 · Mathematics · Linear Algebra · Inner Products and Quadratic Forms

Mathematics (MA) 2021

Let \(\langle \cdot, \cdot \rangle : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}\) be an inner product on the vector space \(\mathbb{R}^n\) over \(\mathbb{R}\). Consider the following statements:
\(P\): \(|\langle u, v \rangle| \leq \frac{1}{2} (\langle u, u \rangle + \langle v, v \rangle)\) for all \(u, v \in \mathbb{R}^n\).
\(Q\): If \(\langle u, v \rangle = \langle 2u, -v \rangle\) for all \(v \in \mathbb{R}^n\), then \(u = 0\).
Then

A

both P and Q are TRUE

B

P is TRUE and Q is FALSE

C

P is FALSE and Q is TRUE

D

both P and Q are FALSE

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14

2023 · Mathematics · Linear Algebra · Inner Products and Quadratic Forms

Mathematics (MA) 2023

Consider \( \mathbb{R}^4 \) with the inner product \( \langle x, y \rangle = \sum_{i=1}^4 x_i y_i \) for \( x = (x_1, x_2, x_3, x_4) \) and \( y = (y_1, y_2, y_3, y_4) \).
Let \( M = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1 = x_3 \} \) and \( M^\perp \) denote the orthogonal complement of \( M \). The dimension of \( M^\perp \) is equal to __________.

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15

2024 · Mathematics · Linear Algebra · Inner Products and Quadratic Forms

Mathematics (MA) 2024

Given a real subspace \(W\) of \(\mathbb{R}^4\), let \(W^\perp\) denote its orthogonal complement with respect to the standard inner product on \(\mathbb{R}^4\). Let \(W_1 = \text{Span}\{(1, 0, 0, -1)\}\) and \(W_2 = \text{Span}\{(2, 1, 0, -1)\}\) be real subspaces of \(\mathbb{R}^4\). The dimension of \(W_1^\perp \cap W_2^\perp\) over \(\mathbb{R}\) is equal to ______ (answer in integer)

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16

2024 · Mathematics · Linear Algebra · Inner Products and Quadratic Forms

Mathematics (MA) 2024

Let \(\langle \cdot, \cdot \rangle\) denote the standard inner product on \(\mathbb{R}^7\). Let \(\Sigma = \{v_1, \ldots, v_5\} \subseteq \mathbb{R}^7\) be a set of unit vectors such that \(\langle v_i, v_j \rangle\) is a non-positive integer for all \(1 \leq i \neq j \leq 5\). Define \(N(\Sigma)\) to be the number of pairs \((r, s), \; 1 \leq r, s \leq 5\), such that \(\langle v_r, v_s \rangle \neq 0\). The maximum possible value of \(N(\Sigma)\) is equal to

A

9

B

10

C

14

D

5

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17

2025 · Mathematics · Linear Algebra · Inner Products and Quadratic Forms

Mathematics (MA) 2025

Consider the inner product space of all real-valued continuous functions defined on \([-1, 1]\) with the inner product \[ \langle f, g \rangle = \int_{-1}^{1} f(x) g(x) \, dx. \] If \(p(x) = \alpha + \beta x^2 - 30x^4\), \(\alpha, \beta \in \mathbb{R}\) is orthogonal to all the polynomials having degree less than or equal to 3, with respect to this inner product, then \(\alpha + 5\beta\) is equal to ______ (in integer)

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18

2007 · Mathematics · Linear Algebra · Inner Products and Quadratic Forms

Mathematics (MA) 2007

Let \(S = \{(0, 1, 1), (1, 0, 1), (-1, 2, 1)\} \subseteq \mathbb{R}^3\). Suppose \(\mathbb{R}^3\) is endowed with the standard inner product \(\langle \cdot, \cdot \rangle\). Define \(M = \{x \in \mathbb{R}^3 : \langle x, y \rangle = 0 \text{ for all } y \in S\}\). Then the dimension of \(M\) equals

A

0

B

1

C

2

D

3

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19

2007 · Mathematics · Linear Algebra · Inner Products and Quadratic Forms

Mathematics (MA) 2007

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

A

\(\{ \frac{1}{\sqrt{3}}(1, 1, 1), \frac{1}{\sqrt{6}}(1, -2, 1) \}\)

B

\((1, 0, 0), (0, 1, 0), (0, 0, 1)\)

C

\(\{ \frac{1}{\sqrt{3}}(1, 1, 1), \frac{1}{\sqrt{2}}(1, -1, 0) \}\)

D

\(\{ \frac{1}{\sqrt{3}}(1, 1, 1), \frac{1}{\sqrt{2}}(0, 1, -1) \}\)

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20

2026 · Mathematics · Linear Algebra · Inner Products and Quadratic Forms

Mathematics (MA) 2026

Let \( P \) be a \( 3 \times 3 \) real symmetric positive definite matrix. Which of the following statements is/are TRUE?

A

\(Q^T P Q \) is positive definite for all nonzero \(Q \in \mathbb{R}^{3 \times 3} \).

B

\(Q^T P Q \) is positive semidefinite for all \(Q \in \mathbb{R}^{3 \times 3} \).

C

\(Q^T P Q \) is positive definite if \(Q \in \mathbb{R}^{3 \times 3} \) is nonsingular.

D

\(Q^T P Q \) is not positive semidefinite for some \(Q \in \mathbb{R}^{3 \times 3} \).

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