[DFL] Prev Year Question

Bangladesh Open University
School of Science and Technology
B. Sc in Computer Science and Engineering Program
152 Term (1st Year 2nd Semester)
Final Examination
Course Code & Title: MAT1231 Linear Algebra and Differential Equation

Time: 3 hours
Total Marks (5×14): 70

[N.B.: Answer any 5 (five) questions. The figures in the right margin indicate the full marks. All portions of each question must be answered sequentially.**]

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1.

(a) Define differential equation. Show that the differential equation of circle touch the x-axis at the origin is

\((x^2 - y^2) dy - 2xy dx = 0\).
1+4

(b) Solve any three of the following equations:
(i) \(\displaystyle \int dy = (y^2 - 1) dx\);
(ii) \(\displaystyle \frac{dy}{dx} = 1 + e^{x - y};\)
(iii) \(\displaystyle \frac{dy}{dx} = \sin(x+y) + \cos(x+y);\)
(iv) \(\displaystyle (x^2 + y^2) dy = xy dx.\)
9

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2.

(a) Prove that the differential equation \(M dx + N dy = 0\) is exact if and only if

\(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x},\)
where \(M\) and \(N\) both are functions of \(x, y\).
6

(b) Solve:
(i) \((x^2 + 3xy^2) dx + (3x^2y + y^3) dy = 0;\)
(ii) \((1 + xy) y dx + (1 - x y) y dy = 0.\)
8

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3.

(a) Find the general solution of \(2y'' - 7y' + 3y = 0.\)
6

(b) It is evident that \(y_p = 3x\) is a particular solution of the equation \(y'' + 4y = 12x\), and that \(y_c(x) = c_1 \cos 2x + c_2 \sin 2x\) is its complementary solution. Find a solution of this differential equation that satisfies the initial conditions \(y(0) = 5, y'(0) = 7\).
4

(c) Show that in a mass-spring-dashpot system, the equation \(mx'' + cx' + kx = 0\), where \(m\) is mass, \(c\) is dashpot constant and \(k\) is spring constant; has a unique solution for \(t \geq 0\) satisfying given initial conditions \(x(0) = x_0, x'(0) = v_0.\)
4

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4.

(a) If \(A = \begin{bmatrix} 2 & -3 & 1 \\ -3 & 1 & 2 \\ 1 & 2 & -3 \end{bmatrix},\) then find the value of \(A^2 - 3A + 9I_3.\)
6

(b) Find the matrix \(X\), such that

\[ \begin{bmatrix} 3 & -4 \\ -3 & -2 \end{bmatrix} X = \begin{bmatrix} -16 & -6 \\ 7 & 2 \end{bmatrix}. \]

4

(c) Solve the following system: 4

\[ \begin{aligned} x + y + z &= 7 \\ x + 2y + 3z &= 16 \\ x + 3y + 4z &= 22. \end{aligned} \]

5.

(a) Transform the following differential equation into an equivalent system of first-order differential equations.

\[ x'' + 3x' + 7x = t^2. \]

7

(b) Find the general solution of the system

\[ x' = y; \\ y' = 2x + y. \]

7

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6.

(a) Show whether the following vectors are a basis of \(\mathbb{R}^3\) or not.

\[ (1, 2, 1), (2, 1, 0), (1, -1, 2). \]

7

(b) Find a basis and the dimension of the solution space \(W\) of the following homogeneous system:

\[ x + 2y + z - 2t = 0 \\ 2x + 4y + 4z - 3t = 0 \\ 3x + 6y + 7z - 4t = 0. \]

4

(c) Define row space, column space and null space of a matrix.
3

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7.

(a) Find the eigenvalues of the matrix

\[ A = \begin{bmatrix} 2 & 1 & 0 \\ 3 & 2 & 0 \\ 0 & 0 & 4 \end{bmatrix} \]

5

(b) Find a matrix \(P\) that diagonalizes the matrix

\[ A = \begin{bmatrix} 0 & 0 & -2 \\ 1 & 2 & 1 \\ 1 & 0 & 3 \end{bmatrix} \]

9

Bangladesh Open University
School of Science and Technology
B. Sc in Computer Science and Engineering Program
162 Term (1st Year 2nd Semester) Final Examination
Course Code & Title: MAT1231 Linear Algebra and Differential Equation

Time: 3 hours
Total Marks: 70

[N.B.: Answer any 5 (five) questions. The figures in the right margin indicate the full marks. All portions of each question must be answered sequentially.**]

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1.

(a) Define a differential equation with examples. Define also order and degree of a differential equation with examples. Distinguish between an ODE and a PDE.
5

(b) Find the differential equations from the following equation:

\[ y = e^x (A \sin 2x + B \cos 2x). \]

5

(c) Solve: \(x^2(1 + y^2) \frac{dy}{dx} + y^2(x - 1)x = 0\).
4

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2.

(a) Define homogeneous and linear differential equation with examples. Solve:

\[ dx + 2y \tan x = \sin x, \quad y\left(\frac{\pi}{3}\right) = 0. \]

2+5

(b) Verify that the differential equation

\[ (1 + e^x) dx + e^x(1 - y^2) dy = 0 \]

is exact and hence solve it.
7

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3.

(a) Solve any two:
(i) \((D^2 - 4D + 13)y = 0\)
(ii) \((D^2 + 4)y = e^{2x}\)
(iii) \((D^2 + 2)^2 y = \cos x\)
8

(b) The particular solution of

\[ x^3 \frac{dy}{dx} + y^3 + \frac{dy}{dx} = 0 \]

when \(y(0) = 0\) and \(y'(0) = 1\).
6

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4.

(a) Define symmetric and skew-symmetric matrices. Prove that every square matrix \(A\) can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix.
1+4

(b) If

\[ A = \begin{bmatrix} 1 & 2 & 1 \\ 2 & 1 & 1 \\ 2 & 2 & 2 \end{bmatrix}, \]

then show that \(A^2 - 4A - 5I = 0\), where \(I, 0\) are the unit matrix and the null matrix of order 3 respectively. Use this result, also find \(A^{-1}\).
3+2

(c) Prove that

\[ A = \begin{bmatrix} 3 & 3 & 3 \\ 3 & 3 & 3 \\ 1 & -5 & 1 \end{bmatrix} \]

is an orthogonal matrix.
4

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5.

(a) Solve the following equation by using inverse of matrix method:

\[ \begin{aligned} 2x - y + 3z &= 9 \\ x + 3y - 2 &= 4 \\ 3x + 2y + 2z &= 10 \end{aligned} \]

5

(b) Find the rank of the matrix,

\[ A = \begin{bmatrix} 1 & 2 & 0 & -1 \\ 2 & 6 & -3 & -3 \\ 13 & 10 & -6 & -5 \end{bmatrix} \]

4

(c) Find the value of the determinant:

\[ \left| \begin{array}{cccc} 2 & -1 & 2 & 4 \\ -1 & 2 & -1 & 0 \\ 1 & -3 & 2 & 1 \\ -4 & -2 & 3 & 2 \end{array} \right| \]

5

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6.

(a) Define a vector space and subspace with example.
4

(b) Show that

\[ S = \{ (a, b, c, d) \in \mathbb{R}^4 : 2a - 3b + 5c - d = 0 \} \]

is a subspace of \(\mathbb{R}^4\).
5

(c) Prove that the vector space \(V\) is the direct sum of its subspaces \(U\) and \(W\) if and only if:
(i) \(V = U + W\);  (ii) \(U \cap W = \{ 0 \}\)
5

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7.

(a) Define eigenvalues and eigenvectors of a square matrix.
3

(b) Find the eigenvalues and the associated eigenvectors of the following matrix:

\[ A = \begin{bmatrix} 8 & 2 & -2 \\ 3 & 3 & -1 \\ 24 & 8 & -6 \end{bmatrix} \]

7

(c) Find the general solution of the following system and then also find a particular solution.

\[ \begin{aligned} x + 2y - 3z &= 6 \\ 2x - y + 4z &= 2 \\ 4x + 3y - 2z &= 14 \end{aligned} \]

5

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Bangladesh Open University
School of Science and Technology
B. Sc in Computer Science and Engineering Program
172 Term (1st Year 2nd Semester) Final Examination
Course Code & Title: MAT1231 Linear Algebra and Differential Equation

Time: 3 hours
Total Marks (5×14): 70

[N.B.: Answer any 5 (five) questions. The figures in the right margin indicate the full marks.
All portions of each question must be answered sequentially.]

---

1.

(a) Define differential equation with examples. Define also order and degree of a differential equation with examples.
2+3 = 5

(b) Find the differential equation whose solution is

\[ y = a + b \ln x + c (\ln x)^2 + 3x^2, \]

where \(a, b,\) and \(c\) are arbitrary constants.
5

(c) Solve:

\[ \sin^{-1} \left(\frac{dy}{dx} \right) = x + y. \]

4

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2.

(a) Define homogeneous differential equation with example. Solve the equation

\[ (6x - 4y + 1) dy = (3x - 2y + 1) dx. \]

1+4 = 5

(b) Determine whether the equation

\[ y \log y \, dx + (x - \log y) \, dy = 0 \]

is exact. If it is, then solve.
4

(c) Define Bernoulli’s equation and hence solve:

\[ \frac{dy}{dx} + \frac{2y}{x} = \frac{y^3}{x^3}. \]

5

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3.

(a) Find the general solution of the following differential equations:
(i) \((D^2 - 3D + 4)y = \cos(4x + 5)\);
(ii) \((D^2 - 6D + 9)y = 1 + x + x^2\);
  (where \(D = \frac{d}{dx}\))
8

(b) Find the particular solution of

\[ \frac{d^2 y}{dx^2} - \frac{dy}{dx} - 6y = 8e^{2x} - 5e^{3x}, \]

when \(y(0) = 3\) and \(y'(0) = 5.\)
6

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4.

(a) Define upper triangular & lower triangular matrix. Show that \((AB)^{-1} = B^{-1} A^{-1}\), where \(A\) and \(B\) are non-singular matrix.
2+3 = 5

(b) Find the adjoint and inverse of the matrix

\[ A = \begin{bmatrix} 2 & -1 & -1 \\ 1 & -2 & 1 \\ 1 & -1 & 2 \end{bmatrix} \]

4

(c) Solve the following linear equations with the help of matrices:

\[ \begin{aligned} x + 2y + 3z + 4 &= 0 \\ 2x + 4y + 5z + 7 &= 0 \\ 3x + 5y + 6z + 10 &= 0 \end{aligned} \]

5

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5.

(a) Define rank of a matrix. Find the rank of matrix

\[ A = \begin{bmatrix} 3 & -2 & 0 & -1 \\ 0 & 2 & 2 & 1 \\ 1 & -2 & 3 & 2 \end{bmatrix} \]

4

(b) Prove that

\[ \left| \begin{matrix} a^2 - bc & b^2 - ca & c^2 - ab \\ c^2 - ab & a^2 - bc & b^2 - ca \\ b^2 - ca & c^2 - ab & a^2 - bc \end{matrix} \right| = (a^3 + b^3 + c^3 - 3abc)^2 \]

5

(c) Using Cramer’s rule solve the followings:

\[ \begin{aligned} x - y + z &= 1 \\ x + y - 2z &= 0 \\ 2x + y - z &= 0 \end{aligned} \]

4

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6.

(a) State and prove second fundamental theorem of subspace.
5

(b) Show that the set of vectors

\[ \{ (3, 0, 1, -1), (2, -1, 0, 1), (1, 1, 1, -2) \} \]

is linearly dependent.
4

(c) State Cayley-Hamilton theorem. Use Cayley-Hamilton theorem to find \(A^{-1}\) of the matrix

\[ A = \begin{bmatrix} 2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{bmatrix} \]

5

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7.

(a) Find the eigenvalues and eigenvectors of the matrix

\[ A = \begin{bmatrix} 4 & 6 & 6 \\ 1 & 3 & 2 \\ -1 & -4 & -3 \end{bmatrix} \]

Also find the matrix \(P\) that diagonalizes \(A\) and determine \(P^{-1} A P\).
7+3

(b) Test whether the transformation defined as follows is linear or not:

\[ T : \mathbb{R}^4 \to \mathbb{R}^3, \quad T(x, y, z, t) = (x - y + z, x + y, y - t) \]

4

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Bangladesh Open University
School of Science and Technology
B. Sc in Computer Science and Engineering Program
182 Term 1st Year 2nd Semester Final Examination
Course Code & Title: MAT1231 Linear Algebra and Differential Equation

Time: 3 hours
Total Marks (5×14): 70

[N.B.: Answer any 5 (five) questions. The figures in the right margin indicate the full marks.
All portions of each question must be answered sequentially.]

---

1.

(a) Define differential equation. Define also order and degree of a differential equation. Solve the differential equation

\[ (1 + x^2) dy - (1 - y^2) dx = 0 \]

2+2+4

(b) Solve any two of the following equations:
(i) \(e^{x - y} dx + e^{y - x} dy = 0\)
(ii) \((x^2 - yx^2) dy + (y^2 + xy^2) dx = 0\)
(iii) \((x^2 + y^2) dy = xy \, dx\)
6

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2.

(a) What is integrating factor of differential equation? Solve the following equations:
(i) \((12y + 4y^3 + 6x^2) dx + 3(x + xy^2) dy = 0\)
(ii) \(x \frac{dy}{dx} + 2y = x^2 \log x\)
1+7

(b) Define homogeneous differential equation with example. Solve the equation:

\[ \frac{dy}{dx} = \frac{y(y + x)}{x(y - x)} \]

2+4

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3.

(a) If

\[ A = \begin{bmatrix} 1 & 3 \\ -2 & 2 \end{bmatrix} \]

then find the matrix \(B\), such that \(AB = I\); where \(I\) is the identity matrix of order 2, also show that matrix \(A\) satisfies the equation

\[ x^2 - 3x + 8 = 0. \]

6

(b) Determine the values of \(\alpha, \beta\) and \(\gamma\), when

\[ \begin{bmatrix} 0 & 2\beta & \gamma \\ \alpha & \beta & -\gamma \\ \alpha & -\beta & \gamma \end{bmatrix} \]

is orthogonal.
4

(c) Prove that

\[ \begin{vmatrix} x & 1 & 1 & \ldots & 1 \\ 1 & x & 1 & \ldots & 1 \\ 1 & 1 & x & \ldots & 1 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & 1 & \ldots & x \end{vmatrix} = (x - 1)^{n - 1} (x + n - 1), \quad \text{where given determinant is of order } n. \]

4

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4.

(a) By using any method determine the value of \(\lambda\) such that the following system of linear equation has
(i) no solution (ii) more than one solution (iii) a unique solution.

\[ \begin{aligned} x + y - z &= 1 \\ 2x + 3y + \lambda z &= 3 \\ x + \lambda y + 3z &= 2 \end{aligned} \]

7

(b) Define rank of a matrix. Find the rank of the following matrix:

\[ A = \begin{bmatrix} 1 & 3 & -2 & 5 & 4 \\ 1 & 4 & 1 & 3 & 5 \\ 1 & 4 & 2 & 4 & 3 \\ 2 & 7 & -3 & 6 & 13 \end{bmatrix} \]

4

(c) Show that

\[ S = \{ (a, b, c, d) \in \mathbb{R}^4 : 2a - 3b + 5c - d = 0 \} \]

is a subspace of \(\mathbb{R}^4\).
3

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5.

(a) Define linear combination. Express the matrix

\[ E = \begin{bmatrix} 5 & 1 \\ -2 & 3 \end{bmatrix} \]

as a linear combination of

\[ A = \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix},\quad B = \begin{bmatrix} 1 & -1 \\ 0 & 1 \end{bmatrix},\quad C = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \]

4

(b) Define vector space and subspace with example.
4

(c) Determine whether the following sets are subspace of \(\mathbb{R}^3\) or not:
(i) \(S = \{ (x, y, 0) : x, y \in \mathbb{R} \}\)
(ii) \(T = \{ (x, y, 1) : x, y \in \mathbb{R} \}\)
6

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6.

(a) Write down the common source of mechanical vibration. Solve the following differential equation:

\[ (3x + 2)^2 \frac{d^2 y}{dx^2} + 5(3x + 2) \frac{dy}{dx} - 3y = x^2 + x + 1 \]

7

(b) A 30V of electromotive force is applied to an LR series circuit in which the inductance is 0.2 henry and resistance is 50 ohms. Find the current \(i(t)\) if \(i(0) = 0\). Determine also the current after a long time.
7

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7.

(a) Define eigenvalues and eigenvectors of a square matrix. Find all the eigenvalues and associated eigenvectors of the following matrix:

\[ A = \begin{bmatrix} 1 & 0 & -2 \\ 0 & 0 & 0 \\ -2 & 0 & 4 \end{bmatrix} \]

2+5

(b) State Cayley Hamilton theorem, verify Cayley Hamilton theorem and hence find \(A^{-1}\) of the following matrix:

\[ A = \begin{bmatrix} 2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{bmatrix} \]

3+4

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Here is the full and clearly formatted written text from both pages of your uploaded exam paper:

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Bangladesh Open University

School of Science and Technology
B.Sc in Computer Science and Engineering Program
202 Term 1st Year 2nd Semester Final Examination
Course Code & Title: MAT1231 Linear Algebra and Differential Equations

---

Time: 3 Hours
Total Marks (5×14): 70

[N.B.: Answer any 5 (five) questions. The figures in the right margin indicate the full marks.
All portions of each question must be answered sequentially.]

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1.

(a) Make the equation homogeneous and find its solution:

\[ \frac{dy}{dx} = \frac{2x + 3y + 3}{2x + 3y + 4} \]

6

(b) Solve the differential equation:

\[ \frac{dy}{dx} = \sin(x + y) + \cos(x + y) \]

4

(c) Solve the equation:

\[ x \sqrt{1 + y^2} dx + y \sqrt{1 + x^2} dy = 0 \]

4

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2.

(a) Prove that the differential equation

\[ M dx + N dy = 0 \]

is exact if and only if

\[ \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \]

6

(b) Verify that the differential equation

\[ y \log y \, dx + (x - \log y) \, dy = 0 \]

is exact or not and hence solve it.
6

(c) Define homogeneous differential equation and solve the equation:

\[ (x + y + 1) dx - (2x + 2y + 1) dy = 0 \]

4

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3.

(a) Define Bernoulli and Riccati equations. Solve:

\[ \frac{dy}{dx} + x \sin 2y = x^2 \cos^2 y \]

2+4

(b) Find the particular solution of

\[ (D^2 + 4)y = 12x,\quad \text{when } y(0) = 5,\; y'(0) = 7 \]

4

(c) A 30 volt electromotive force is applied to an LR series circuit in which the inductance is 0.2 henry and the resistance is 50 ohms. Find the current \(i(t)\) if \(i(0) = 0\). Determine also the current after a long time.
4

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4.

(a) Define symmetric and skew-symmetric matrix. If

\[ A = \begin{bmatrix} 2 & -3 \\ 4 & 1 \end{bmatrix} \]

then prove that \(A\) can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix.
4

(b) Let

\[ A = \begin{bmatrix} 2 & 2 & 0 \\ 2 & 2 & 1 \\ -7 & 2 & -3 \end{bmatrix} \]

be a matrix, is \(A^{-1}\) exists? If exists, find \(A^{-1}\) and establish that \(AA^{-1} = I\).
6

(c) Write down two matrices \(A\) and \(B\) of order 3×3 and then prove that

\[ (AB)^T = B^T A^T \]

4

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5.

(a) Define rank of a matrix. Find the rank of matrix

\[ A = \begin{bmatrix} 3 & -2 & 0 & -1 \\ 0 & 2 & 2 & 1 \\ 1 & -2 & 3 & 2 \\ 0 & 1 & 2 & 1 \end{bmatrix} \]

4

(b) Using Cramer’s Rule, solve the following system of equations:

\[ \begin{aligned} x - y + z &= 1 \\ x + y - 2z &= 0 \\ 2x - y - z &= 0 \end{aligned} \]

5

(c) Determine the value of \(k\) such that the following system of linear equations has:
(i) no solution (ii) more than one solution (iii) a unique solution

\[ \begin{aligned} x + y + kz &= 1 \\ x + ky + z &= 1 \\ kx + y + z &= 1 \end{aligned} \]

5

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6.

(a) State and prove Cayley-Hamilton theorem.
4

(b) Determine Eigenvalues and Eigenvectors for the matrix

\[ B = \begin{bmatrix} 1 & 2 \\ 4 & 3 \end{bmatrix} \]

6

(c) Write a short note on Population Modeling and Equilibrium Solution.
4

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7.

(a) Define linear dependence and linear independence. Test whether the following vectors are linearly independent or dependent:

\[ (1, -2, 4, 1),\; (2, 1, 0, -3),\; (3, -6, 1, 4) \]

5

(b) Define subspace. Show that

\[ S = \{ (a, b, c, d) \in \mathbb{R}^4 : 2a - 3b + 5c - d = 0 \} \]

is a subspace of \(\mathbb{R}^4\).
1+4

(c) Find matrix representation of \(T\) for the given basis

\[ f_1 = (1, 1, 0),\; f_2 = (1, 0, 1),\; f_3 = (0, 1, 1) \]

where \(T: \mathbb{R}^3 \to \mathbb{R}^3\), defined by

\[ T(x, y, z) = (x + y, y + z, z + x) \]

4

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