Vector Spaces and Linear Transformations - Linear Algebra - Mathematics Previous Year Questions

1

2008 · Mathematics · Linear Algebra · Vector Spaces and Linear Transformations

Mathematics (MA) 2008

Consider the subspace \( W = \{[a_{ij}] : a_{ij} = 0 \text{ if } i \text{ is even}\} \) of all \(10 \times 10\) real matrices. Then the dimension of \(W\) is

A

25

B

50

C

75

D

100

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2

2008 · Mathematics · Linear Algebra · Vector Spaces and Linear Transformations

Mathematics (MA) 2008

Let \(T: \mathbb{R}^4 \to \mathbb{R}^3\) be the linear map satisfying \(T(e_1) = e_1, T(e_2) = e_3, T(e_3) = 0, T(e_4) = e_3\), where \(\{e_1, e_2, e_3, e_4\}\) is the standard basis of \(\mathbb{R}^4\). Then

A

\(T\) is idempotent

B

\(T\) is invertible

C

Rank \(T = 3\)

D

\(T\) is nilpotent

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3

2008 · Mathematics · Linear Algebra · Vector Spaces and Linear Transformations

Mathematics (MA) 2008

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

A

\(\left\{(1,0,0)^T, \left(0, \frac{2}{\sqrt{5}}, \frac{1}{\sqrt{5}}\right)^T, \left(\frac{2}{\sqrt{6}}, \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}}\right)^T\}\)

B

\(\left\{(1,0,0)^T, \left(0, \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)^T\}\)

C

\(\left\{(1,0,0)^T, \left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)^T, \left(\frac{2}{\sqrt{6}}, \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}}\right)^T\}\)

D

\(\left\{(1,0,0)^T, (0,0,1)^T\}\)

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4

2008 · Mathematics · Linear Algebra · Vector Spaces and Linear Transformations

Mathematics (MA) 2008

For any \(n \in \mathbb{N}\), let \(P_n\) denote the vector space of all polynomials with real coefficients and of degree at most \(n\). Define \(T: P_n \to P_{n+1}\) by \(T(p)(x) = p'(x) - \int_0^1 p(t) dt\). Then the dimension of the null space of \(T\) is

A

0

B

1

C

\(n\)

D

\(n+1\)

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5

2008 · Mathematics · Linear Algebra · Vector Spaces and Linear Transformations

Mathematics (MA) 2008

If \(\begin{pmatrix} 1 & 4 & 3 \\ 2 & 7 & 9 \\ 5 & 8 & a \end{pmatrix} = \begin{pmatrix} l_{11} & 0 & 0 \\ l_{21} & l_{22} & 0 \\ l_{31} & l_{32} & -53 \end{pmatrix} \begin{pmatrix} 1 & u_{12} & u_{13} \\ 0 & 1 & u_{23} \\ 0 & 0 & 1 \end{pmatrix}\), then the value of \(a\) is

A

-2

B

-1

C

1

D

2

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6

2008 · Mathematics · Linear Algebra · Vector Spaces and Linear Transformations

Mathematics (MA) 2008

If \(M\) is any \(3 \times 3\) real matrix, then trace \((NMN^T)\) is equal to

A

\([\text{trace}(N)]^2 \text{trace}(M)\)

B

\(2 \text{trace}(N) + \text{trace}(M)\)

C

\(\text{trace}(M)\)

D

\([\text{trace}(N)]^2 + \text{trace}(M)\)

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7

2009 · Mathematics · Linear Algebra · Vector Spaces and Linear Transformations

Mathematics (MA) 2009

The dimension of the vector space V = {A = (a_ij)_n×n : a_ij ∈ ℂ, a_ij = –a_ji} over the field ℝ is

A

n²

B

n² – 1

C

n² – n

D

n²/2

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8

2009 · Mathematics · Linear Algebra · Vector Spaces and Linear Transformations

Mathematics (MA) 2009

The dimension of the range space of \( T^2 \) is

A

0

B

1

C

2

D

3

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9

2010 · Mathematics · Linear Algebra · Vector Spaces and Linear Transformations

Mathematics (MA) 2010

If the nullity of the matrix \( \begin{bmatrix} k & 1 & 2 \\ 1 & -1 & -2 \\ 1 & 1 & 4 \end{bmatrix} \) is 1, then the value of \( k \) is

A

-1

B

0

C

1

D

2

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10

2010 · Mathematics · Linear Algebra · Vector Spaces and Linear Transformations

Mathematics (MA) 2010

Let \( T : P_3[0, 1] \to P_2[0, 1] \) be defined by \( (Tp)(x) = p''(x) + p'(x) \). Then the matrix representation of \( T \) with respect to the bases \( \{1, x, x^2, x^3\} \) and \( \{1, x, x^2\} \) of \( P_3[0, 1] \) and \( P_2[0, 1] \) respectively is

A

\(\begin{bmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 2 & 2 & 0 \\ 0 & 6 & 3 \end{bmatrix} \)

B

\(\begin{bmatrix} 0 & 1 & 2 & 0 \\ 0 & 0 & 2 & 6 \\ 0 & 0 & 0 & 3 \end{bmatrix} \)

C

\(\begin{bmatrix} 0 & 2 & 1 & 0 \\ 6 & 2 & 0 & 0 \\ 3 & 0 & 0 & 0 \end{bmatrix} \)

D

\(\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 2 & 2 \\ 3 & 6 & 0 \end{bmatrix} \)

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11

2010 · Mathematics · Linear Algebra · Vector Spaces and Linear Transformations

Mathematics (MA) 2010

Consider the basis \( \{u_1, u_2, u_3\} \) of \( \mathbb{R}^3 \), where \( u_1 = (1, 0, 0) \), \( u_2 = (1, 1, 0) \), \( u_3 = (1, 1, 1) \). Let \( \{f_1, f_2, f_3\} \) be the dual basis of \( \{u_1, u_2, u_3\} \) and \( f \) be a linear functional defined by \( f(a, b, c) = a + b + c \), \( (a, b, c) \in \mathbb{R}^3 \). 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)

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12

2011 · Mathematics · Linear Algebra · Vector Spaces and Linear Transformations

Mathematics (MA) 2011

The solution \( z = [z_1, z_2, z_3]^T \) of the system \( L z = b \) is

A

[-1, -1, -2]^T

B

[1, -1, 2]^T

C

[1, -1, -2]^T

D

[-1, 1, 2]^T

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13

2011 · Mathematics · Linear Algebra · Vector Spaces and Linear Transformations

Mathematics (MA) 2011

The solution \( x = [x_1, x_2, x_3]^T \) of the system \( U x = z \) is

A

[2, 1, -2]^T

B

[2, 1, 2]^T

C

[-2, -1, -2]^T

D

[-2, 1, -2]^T

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14

2012 · Mathematics · Linear Algebra · Vector Spaces and Linear Transformations

Mathematics (MA) 2012

Let \(V=\mathbb{C}^2\) be the vector space over the field of complex numbers and \(B=\{(1,i),(i,1)\}\) be a given ordered basis of \(V\). Then for which of the following, \(B^*=\{f_1, f_2\}\) is a dual basis of \(B\) over \(\mathbb{C}\)?

A

\(f_1(z_1,z_2)=\frac{1}{2}(z_1-iz_2),\ f_2(z_1,z_2)=\frac{1}{2}(z_1+iz_2)\)

B

\(f_1(z_1,z_2)=\frac{1}{2}(z_1+iz_2),\ f_2(z_1,z_2)=\frac{1}{2}(iz_1+z_2)\)

C

\(f_1(z_1,z_2)=\frac{1}{2}(z_1-iz_2),\ f_2(z_1,z_2)=\frac{1}{2}(-iz_1+z_2)\)

D

\(f_1(z_1,z_2)=\frac{1}{2}(z_1+iz_2),\ f_2(z_1,z_2)=\frac{1}{2}(-iz_1-z_2)\)

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15

2012 · Mathematics · Linear Algebra · Vector Spaces and Linear Transformations

Mathematics (MA) 2012

Let the linear transformation \(T:F^2\rightarrow F^3\) be defined by \(T(x_1,x_2)=(x_1,x_1+x_2,x_2)\). Then the nullity of \(T\) is

A

0

B

1

C

2

D

3

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16

2012 · Mathematics · Linear Algebra · Vector Spaces and Linear Transformations

Mathematics (MA) 2012

Let \( T: P_{3} \rightarrow P_{3} \) be the map given by \( T(p(x)) = \int_{0}^{x} p'(t) dt \). If the matrix of \( T \) relative to the standard bases \( B_{1} = B_{2} = \{1, x, x^{2}, x^{3}\} \) is \( M \) and \( M' \) denotes the transpose of the matrix \( M \), then \( M + M' \) is

A

\(\begin{pmatrix} 0 & -1 & -1 & -1 \\ -1 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{pmatrix} \)

B

\(\begin{pmatrix} -1 & 0 & 0 & 2 \\ 0 & -1 & 1 & 0 \\ 0 & 1 & -1 & 0 \\ 2 & 0 & 2 & -1 \end{pmatrix} \)

C

\(\begin{pmatrix} 2 & 0 & 0 & -1 \\ 0 & 2 & 1 & 0 \\ 0 & 1 & 2 & -1 \\ -1 & 0 & -1 & 0 \end{pmatrix} \)

D

\(\begin{pmatrix} 0 & 2 & 2 & 2 \\ 2 & -1 & 0 & 0 \\ 2 & 0 & -1 & 0 \\ 2 & 0 & 0 & -1 \end{pmatrix} \)

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17

2013 · Mathematics · Linear Algebra · Vector Spaces and Linear Transformations

Mathematics (MA) 2013

Let \(V\) be the real vector space of all polynomials in one variable with real coefficients and having degree at most 20. Define the subspaces \(W_1 = \{p \in V : p(1) = 0, p\left(\frac{1}{2}\right) = 0, p(5) = 0, p(7) = 0\}\) and \(W_2 = \{p \in V : p\left(\frac{1}{2}\right) = 0, p(3) = 0, p(4) = 0, p(7) = 0\}\). Then the dimension of \(W_1 \cap W_2\) is ______

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18

2013 · Mathematics · Linear Algebra · Vector Spaces and Linear Transformations

Mathematics (MA) 2013

Let \(M\) be the real vector space of \(2 \times 3\) matrices with real entries. Let \(T: M \to M\) be defined by \(T\left(\begin{pmatrix} x_1 & x_2 & x_3 \\ x_4 & x_5 & x_6 \end{pmatrix}\right) = \begin{pmatrix} -x_6 & x_4 & x_1 \\ x_3 & x_5 & x_2 \end{pmatrix}\). The determinant of \(T\) is ______

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19

2013 · Mathematics · Linear Algebra · Vector Spaces and Linear Transformations

Mathematics (MA) 2013

The matrix \(A = \begin{bmatrix} 1 & 2 & 0 \\ 1 & 3 & 1 \\ 0 & 1 & 3 \end{bmatrix}\) can be decomposed uniquely into the product \(A = LU\), where \(L = \begin{bmatrix} 1 & 0 & 0 \\ l_{21} & 1 & 0 \\ l_{31} & l_{32} & 1 \end{bmatrix}\) and \(U = \begin{bmatrix} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \end{bmatrix}\). The solution of the system \(LX = [1 \ 2 \ 2]^t\) is

A

\([1 \ 1 \ 1]^t\)

B

\([1 \ 1 \ 0]^t\)

C

\([0 \ 1 \ 1]^t\)

D

\([1 \ 0 \ 1]^t\)

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20

2013 · Mathematics · Linear Algebra · Vector Spaces and Linear Transformations

Mathematics (MA) 2013

Let \(M\) be the space of all \(4 \times 3\) matrices with entries in the finite field of three elements. Then the number of matrices of rank three in \(M\) is

A

\((3^4 - 1)(3^4 - 3^2)(3^4 - 3^3)\)

B

\((3^4 - 1)(3^4 - 2)(3^4 - 3)\)

C

\((3^4 - 1)(3^4 - 3)(3^4 - 3^2)\)

D

\(3^4(3^4 - 1)(3^4 - 2)\)

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