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VL2021220100210 DA (1)

Vellore Institute of Technology
Department of Mathematics
Digital Assignment for MAT3004
Ankush Chanda
November 17, 2021
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1. Let S = {x1 , x2 , . . . , xk } be linearly independent vectors in a vector space.
A. If we add a vector xk+1 to the collection, will we still have a linearly independent
collection of vectors? Explain with an example.
B. If we delete a vector, say xk , from the collection, will we still have a linearly independent collection of vectors? Explain with an example.
2. Let S be the subspace of P3 (R) consisting of all polynomials p(x) such that p(0) = 0, and
let T be the subspace of all polynomials q(x) such that q(1) = 0. Find bases for S, T and
S ∩ T.
3. Determine whether the following are linear transformations from C[0, 1] onto R:
A. L(f ) =
B. L(f ) =
[f (0)+f (1)]
o 12
[f (x)] dx
4. Find the standard matrix representation for each of the following linear operators:
A. L is the linear operator that reflects each vector x in about the x1 axis and then
rotates it 90◦ in the counter-clockwise direction.
B. L is the linear operator that rotates each x in by 45◦ in the clockwise direction.
5. Let W be the subspace of R4 spanned by w1 = (2, 0, 3, −4), w2 = (4, 2, −5, 1), w3 =
(2, −2, 14, −13), w4 = (6, 2, −2, −3). Is W = R4 ? If not, find a basis of W and extend
it to a basis of R4 .