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Department of Mathematics and Statistics

Fall 2021 MAT 2141A

Assignment 1

1. Let S = {a, b} be a set consisting of two distinct elements. We define operations ‘+’ and ‘·‘ on S as follows:

Determine whether S with respect to ‘+’ and ’·’ is a field or not. Justify your answer.

2. Prove that the function  defined by is surjective.

3. Let V = F × F = {(a1, a2) | a1, a2 ∈ F}, where F is a field. Define addition of elements of V as (a1, a2) + (b1, b2) = (a1 + b2, a2 · b1) and the scalar multiplication as c(a1, a2) = (ca1, ca2). Is V a vector space over F? Justify your answer.

4. Let S = {0, 19}. Consider the vector space V = F(S, R) of functions from S to R.

Prove that h = f − g in V , where f, g, h are ‘vectors’ in V defined by

5. Let U and W be subspaces of a R-vector space V . Using the subspace test prove that the subset

is a subspace of V .

6. Determine whether the following subset is a subspace of  under the operations of addition and scalar multiplication defined on . Justify your answer.

7. Consider the vector space . Determine whether the given vector is in the span of the subset S. Justify your answer.

8. (a) Provide an example of two non-zero distinct vectors f and g in the vector space of real-valued functions F(R, R) such that    (1pt)

(b) Explain why the equality holds for your example.    (2pt)