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GTU First Year Engineering (Semester 2)
Vector Calculus and Linear Algebra
May 2013
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1 (a) (i) Is the matrix $$\begin{bmatrix}1 &0 &4 &6 \\0 &1 &5 &7 \\0 &0 &0 &1 \end{bmatrix}$$in now echelon form or reduced row-echelon form?
2 M
1 (a) (ii) Find the rank of the matrix $\begin{bmatrix}8 &0 &0 &16 \\0 &0 &0 &6 \\0 &9 &9 &9 \end{bmatrix}$
2 M
1 (b) (i) Find the inverse of the matrix$\begin{bmatrix}1&2 \\ 1 & 3 \end{bmatrix}$
2 M
1 (b) (ii) Find the Eigen values of the matrix AT. Is A an invertible? $A= \begin{bmatrix}1 & 3 & 4 \\ 0& 0& 5 \\ 0&0&9 \end{bmatrix}$
2 M
1 (c) (i) Express the following quadratic forms in matrix notation.
i) x2-4xy+y2
ii) 2x2+6xy-5y2
3 M
1 (c) (ii) If ϕ=3x2y-y3z2, find grad ϕ at the point (1, -2, -1).
3 M

2 (a) (i) Find the length of the arc of the curve y=log sec x from $x=0 \ to \ x= \dfrac{\pi}{3}$
3 M
2 (a) (ii) Prove that $\iint_S \overrightarrow{F} \cdot \widehat{n}dS = 3v , \ if \ \overrightarrow{F} = x\widehat{i} + y \widehat{j}+ z \widehat{k}$ where S is any closed surface enclosing volume V.
3 M
2 (b) $If \ \overrightarrow{F}= 3xy \widehat{i} - y^2 \widehat{j}; \ evaluate \ \int_c \overrightarrow{F}\cdot \overrightarrow{d}r.$ Where C is the arc of the Parabola y=2x2 from (0,0) to (1,2).
4 M
2 (c) Show that $\overrightarrow{F} = (y^2 - z^2 + 3yz - 2x)\widehat{i} + (3xz + 2xy)\widehat{j}+ (3xy-2xz + 2z)\widehat{k}$ is both solenoidal and irrotational.
4 M

3 (a) Solve the following homogeneous system of equations by using Gauss-Jorfan elimination
2x1+2x2-x3+x5=0,
-x1-x2+2x3-3x4+x5=0,
x1+x2-2x3-x5=0,
x3+x4+x5=0
5 M
3 (b) Use Cramer's rule to solve:-
x+5y=2
11x+y+2z=3
x+5y+2z=1
5 M
3 (c) What conditions must b1, b2, b3 satisfy in order for the system of equations to be consistent?
x1+2x+3x=b1
2x1+5x2+3x3=b2
x1+ 8x3=b3
4 M

4 (a) Show that the set of all 2×2 matrices of the form $\begin{bmatrix}a &1 \\1 &b \end{bmatrix}$ with addition defined by $\begin{bmatrix}a &1 \\ 1 &b \end{bmatrix} + \begin{bmatrix}c &1 \\ 1 &b \end{bmatrix} = \begin{bmatrix}a+c &1 \\ 1 &b+d \end{bmatrix}$ and scalar multiplication defined by $\begin{bmatrix}a&1 \\ 1 &b \end{bmatrix} = \begin{bmatrix}ka&1 \\ 1 &kb \end{bmatrix}$ is a vector space.
6 M
4 (b) Determine which of the following are subspaces of R3.
(1) all the vector of the form (a,b,c) where b=a+c.
(2) all the vector of the form (a,b,c) where b=a+c+1.
4 M
4 (c) Let v1=1-3x+2x2, v2=1-x+4x2, v3=1-7x. Show that the set S= {v1, v2, v3} is a basis for P2.
4 M

5 (a) Use the inner product $(f,g)= \int^1_0 f(x) g(x) dx;$ to compute d(f,g) and ||f||. Where f(x) = x, g(x)=ex in C[0,1].
4 M
5 (b) Let R2 have the Euclidean inner product. Use the Gram Schmidt process to transform the basis {u1, u2, u3} into an orthonormal basis u1=(1,1,1), u2=(-1,1,0), u3=(1,2,1)
5 M
5 (c) Verify that the vectors u1=(1,-1,2,-1) u2=(-2,2,3,2), u3=(1,2,0,-1), u4=(1,0,0,1) form an Orthogonal basis for R4 with the Euclidean inner product. Also convert it to an orthonormal set by normalizing the vectors.
5 M

6 (a) Find the eigen values of the eigenspaces of the matrix $\begin{bmatrix}0 &0 &-2 \\1 &2 &1 \\1 &0 &3 \end{bmatrix}$
5 M
6 (b) Find matrix P that diagonalizes $A = \begin{bmatrix}1&0 \\ 6 & -1 \end{bmatrix} .$ Also determine P-1 AP.
5 M
6 (c) Using Cayley Hamilton theorem, find A-1; $A= \begin{bmatrix}1 & 4 \\ 2&3\end{bmatrix}$
4 M

7 (a) $Let \ T_A: R^6 \to R^4 \ be \ multiplication \ by \\ A= \begin{bmatrix}-1 &2 &0 &4 &5 &-3 \\3 &-7 &2 &0 &1 &4 \\2 &-5 &2 &4 &6 &1 \\4 &-9 &2 &-4 &-4 &7 \end{bmatrix}$ Find the rank and nullity of TA
5 M
7 (b) Consider the basis S={v1, v2} for R2, where v1=(-2,1) and v2=(-1,3). Let T:R2→R3, be the linear operator such that T(v1)= (-1, 2, 0) and T(v2)=(0, -3, 5) find the formula T(x1, x2), and use that formula to find T(2, -3).
5 M
7 (c) Show that the following functions are linear transformations.
i) T:R2 → R2, where T(x,y)=(x+3y, 3x-y)
ii) T:R2 → R2, where T(x,y)=(2x-y, x-y)
4 M

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