STUPIDSID
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
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[123253108]
3 M
1 (b) (ii)
Solve the system of equation
-2b+3c=1
3a+6b-3c=-2
6a+6b+3c=5
by Gaussian elimination
-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=[1−1203100−1240] 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
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=[−12045−33−720142−524614−92−4−47]
5 M
Attempt the following:
2 (b) (i)
If verctor r=xˆi=yˆi+zˆk then show that ∇rn=nrn−2 (vector r)
2 M
2 (b) (ii)
Prove that ∇2f(r)=f′(r)+2rf(r)
3 M
Attempt the following:
2 (c) (i)
Show that f1=1, f2=ex, f3=e2x, from a linearly, independent set of vectors in C2(-infty, \infty)
2 M
2 (c) (ii)
Check whether the set W={a0+a1x+a2x2+a3x3 where]a0+a1+a2+a3=0, ai ϵ R} is subspace of p3.
2 M
3 (a)
Show that the set of all 2 × 2 matrices of the form [a11b] with addition defined by [a11b]+[c11b]=[a+c11b+d] and scalar multiplicationk[a11b]=[ka11kb] 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={p1,p2,p3}where p=2−x+x2, p1=1+x, p2=1+x2, p3=x+x3
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.
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)=∫1−1p(x)q(x)dx i) Find ||p|| for p=x2
ii) Find d(p,q) if p=1 and q=x
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=[1371101/23800040002]
2 M
4 (c) (ii)
Find k, 1 and m to make A, a Hemition matrix A=[−1k−i3−5i0ml2+4i2]
2 M
5 (a)
Find bases for the eiganspace of A=[002121103]
4 M
Attempt the following:
5 (b) (i)
Use Cayley Hamilton theorem of find A-1 for A=[137423121]
3 M
5 (b) (ii)
Find a matrix P that diagonalize A=[00−2121103]
3 M
5 (c)
Find a change of variable that will reduce the quadratic form x21−x23−4x1x2+4x2x3 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)=(3√x,3√y)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([x1x2])=[x2−5x1+13x2−7x1+16x2]. Find the matrix for the transformation T with respect to the bases B={u1,u2} for R2 and B′={v1,v2,v3} for R3, where u1=[31], u2[52], v1=[10−1], v2=[−122], v3=[012]
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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