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GTU First Year Engineering (Semester 2)
Vector Calculus and Linear Algebra
May 2012
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) Solve the following homogeneous system of linear eq uation by using Gauss Jordan elimination.
2x1+2x2-x3+x5=0
-x1-x2+2x3-3x4+x5=0
x1+x2-2x3-x5=0
x3+x4+x3=0
4 M
Attempt the following:
1 (b) (i) Find A-1 using row operation if $A\begin{bmatrix}1 &2 &3 \\2 &5 &3 \\1 &0 &8 \end{bmatrix}$
3 M
1 (b) (ii) Solve the system of equation
-2b+3c=1
3a+6b-3c=-2
6a+6b+3c=5
by Gaussian elimination
3 M
Attempt the following:
1 (c) (i) "Find the rank of the matrix $A = \begin{bmatrix}1 &-1 &2 &0 \\3 &1 &0 &0 \\-1 &2 &4 &0 \end{bmatrix}$ in terms of determinates"
2 M
1 (c) (ii) Use Cranner's rule to solve
x+2y+z=5
3x-y+z=6
x+y+4z=7
2 M

2 (a) Define the rank and nullity. Find the rank and nullity of the matrix $A=\begin{bmatrix}-1 &2 &0 &4 &5 &-3 \\3 &-7 &2 &0 &1 &4 \\2 &-5 &2 &4 &6 &1 \\4 &-9 &2 &-4 &-4 &7 \end{bmatrix}$
5 M
Attempt the following:
2 (b) (i) $If \ verctor \ r=x\widehat{i} = y \widehat{i} + z \widehat{k} \ then \ show \ that \ \nabla r^n = nr^{n-2} \ (vector \ r)$
2 M
2 (b) (ii) $Prove \ that \ \nabla^2 f (r) = f'(r) + \dfrac {2}{r} f(r)$
3 M
Attempt the following:
2 (c) (i) Show that $f_1 = 1 , \ f_2= e^x, \ f_3 = e^{2x},$ from a linearly, independent set of vectors in C2(-infty, \infty)
2 M
2 (c) (ii) Check whether the set $W= \left \{ a_0 + a_1 x + a_2 x^2 + a_3 x^3 \ where ] a_0 + a_1 + a_2 + a_3 = 0, \ a_i \ \epsilon \ R \right \}$ is subspace of p3.
2 M

3 (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$k \begin{bmatrix}a &1 \\1 &b \end{bmatrix} = \begin{bmatrix}ka &1 \\1 &kb \end{bmatrix}$ is a vector space.
5 M
Attempt the following:
3 (b) (i) Find a standard basis vector that can be added to the set S={(1,0,3), (2,1,4)} to produce a basis of R3
3 M
3 (b) (ii) Find the co-ordinate vector of P relative to the basis $S= \{ p_1, p_2 , p_3\} \\ where \ p=2-x+x^2, \ p_1 = 1 + x, \ p_2 =1+x^2, \ p_3 = x +x^3$
2 M
Attempt the following:
3 (c) (i) Find two vector in R2 with Euclidean Norm 1 whose inner product with (-3,1) is zero.
2 M
3 (c) (ii) If v1, v2, v3 ....... vr are pairwise orthogonal vectors in Rn then ||v1+v2+........ + vr||2 = ||v1||2 + ||v2||2 + .......... + ||vr||2
2 M

4 (a) Let R2 have the Euclidean inner product. Use the Gram Schmidt process to transform the basis {u1, u2, u3} into an orthonormal basis u1=(1,0,0), u2=(3,7,-2), u3=(0,4,1)
4 M
Attempt the following:
4 (b) (i) Find the least squares solution of the linear system Ax=B given by
x1-x2=4
3x1+2x=1
-2x1+4x=3
and find the orthogonal Projection of B on the column space of A.
3 M
4 (b) (ii) Let the vector space p2 have the inner product $(p,q)= \int^1_{-1} p(x)q(x)dx$ i) Find ||p|| for p=x2
ii) Find d(p,q) if p=1 and q=x
3 M
Attempt the following:
4 (c) (i) Define the eiganvalue and eiganvector. Find the eigen value of A9 for $A= \begin{bmatrix} 1 &3 &7 &11 \\0 &1/2 &3 &8 \\0 &0 &0 &4 \\0 &0 &0 &2\end{bmatrix}$
2 M
4 (c) (ii) Find k, 1 and m to make A, a Hemition matrix $A=\begin{bmatrix} -1 &k &-i \\3-5i &0 &m \\l &2+4i &2 \end{bmatrix}$
2 M

5 (a) Find bases for the eiganspace of $A= \begin{bmatrix} 0 &0 &2 \\1 &2 &1 \\1 &0 &3 \end{bmatrix}$
4 M
Attempt the following:
5 (b) (i) Use Cayley Hamilton theorem of find A-1 for $A= \begin{bmatrix} 1&3&7 \\ 4&2&3 \\ 1&2&1\end{bmatrix}$
3 M
5 (b) (ii) Find a matrix P that diagonalize $A = \begin{bmatrix}0&0&-2 \\ 1&2&1 \\ 1&0&3 \end{bmatrix}$
3 M
5 (c) Find a change of variable that will reduce the quadratic form $x^2_1-x^2_3-4_{x_1 x_2}+ 4_{x_2x_3}$ to a sum of squares and express the quadratic form in terms of the new variable.
4 M

6 (a) Verify Green's theorem for the field f(x,y)=(x-y)i+xj and the region R bounded by the unit circle C:r(t)=(cost)i+ (sint)j, 0≤t≤2π
4 M
Attempt the following:
6 (b) (i) Find the flux F=4xzi-y^2j+yzk outward through the surface of the cube cut from the first octant by the planes x=1, y=1 and z=1
3 M
6 (b) (ii) Determine whether T:R2→R2 is linear operator $1) \ T(x,y)= \left(\sqrt[3]{x}, \sqrt[3]{y} \right ) \\ 2) \ T(x,y) = (x,0)$
3 M
Attempt the following:
6 (c) (i) Find the derivative of f(x,y)=x2sin2y at the point (1, π/2) in the direction of v=3i-4j
2 M
6 (c) (ii) Use matrix multiplication to find the image of the vector (-2, 1, 2) if it is rotated -45° about the y-axis.
2 M

7 (a) Verify Stoke's Theorem for the hemisphere S:x2+y2+z2=9, z≥0 its bounding circle C:x2+y2=9, z=0 and the field F=yi-xj
4 M
Attempt the following:
7 (b) (i) Consider the basis S={v1, v2, v3} for R3, v1=(1,2,1), v2=(2,9,0) and v3=(3,3,4) and let T:R3→R2 be the linear transform such that T(v1)=(1,0), T(v2)=(-1,1), T(v3)=(0,1). Find a formula for T(x1, x2, x3) and use that formula to find T(7, 13, 7)
3 M
7 (b) (ii) Let T:R2→R3 be the linear transformation defined by $T \left ( \begin{bmatrix}x_1\\x_2\end{bmatrix} \right )= \begin{bmatrix}x_2\\-5x_1+13x_2 \\-7x_1 + 16x_2 \end{bmatrix} .$ Find the matrix for the transformation T with respect to the bases $B=\{{u}_1, {u}_2 \}\ for \ R^2 \ and \ B' = \{ {v}_1, {v}_2, {v}_3 \}\ for \ R^3, \\ \ where \ {u}_1 = \begin{bmatrix}3 \\1 \end{bmatrix} , \ {u}_2\begin{bmatrix}5 \\2\end{bmatrix}, \ {v}_1 = \begin{bmatrix}1 \\ 0 \\-1 \end{bmatrix} , \ {v}_2 = \begin{bmatrix}-1 \\ 2 \\ 2 \end{bmatrix}, \ {v}_3 = \begin{bmatrix}0\\ 1 \\2\end{bmatrix}$
3 M
Attempt the following:
7 (c) (i) Determine whether the linear transformation T:R2→R2, where T(x,y)=(x,y,x+y) is one
2 M
7 (c) (ii) Find the standard matrix for the linear operator on R2, an orthogonal projection on the y-axis followed by a contraction with factor k=1/3
2 M

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