Sybsc Tutorial No 8

SYBSC TUTORIAL No. Paper II BASIS, Q1. a) The vector u1= (1,2) and u2 = (4,-7) form a basis of ℝ2. Find the coordinates a, b ∈ ℝ of v = (5,3) relative to u1, u2. b) S ={t3+t2, t2+t,t+1,1} is a basis of P3(t). Find the coordinates a, b, c, d ∈ ℝ of v = 2t3+t2-4t+2 relative to S. Q2. Find a basis and dimension of following subspaces. i)

U = {(a, b, c, d) | b – 2c +d = 0 }

ii)

W = {(a, b, c, d) | a = d, b = 2c}

iii)

U∩W

iv)

X = {(a, b, c, d) | a = 2b = 3c}

Q3. Determine whether the following sets are linearly independent. i)

{t3-4t2+3t+3, t3+2t2+4t-1, 2t3- t2-3t+5} in P3(t).

ii)

{et, sin t, t2 } in C(ℝ).

iii)

1  0

−1  2 0 −1 , ,  2  −1 3  0

0  3 , − 2  − 2

− 3 5 , 2   3

4  in M2X2(ℝ). 2 

Q4. Find a subset u1, u 2, u3, u4 that gives a basis for W = L({u1, u 2, u3, u4 }) where u1 =

1 1 

1 1 , u

2=

1 1 

−1 0  , u

3=

1 0 

1 0 , u

4=

1 0 

0 0 .

Q5. Check whether following sets are bases of V. Justify your answer. a) S1 = { (1, 1), (2 , 2)} , S2 = { (1, 1), (1 , 0), ( 0, 1)}, S3 = { (1, 0)} where V= ℝ2. b) S1 = { (1, 0, 0), (2 , 0, 0), (0, 1, 0)} , S2 = { (1, 2, 3), (1, -1, 2)} S3 = { (1, 1, 1), (1 , 0, 0), ( 0, 1, 2), (2 , 1, 3)}

where V= ℝ3.