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MA 1111: Linear Algebra I Tutorial problems, November 25, 2015

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MA 1111: Linear Algebra I Tutorial problems, November 25, 2015

1. For each of the following subsets of the set of all polynomials in one variable find out whether this subset is a subspace.

(a) all polynomials fof degree less than 100 such thatf(1) =0;

(b) all polynomials fof degree less than 100 such that f(1) =f(2) =0;

(c) all polynomials fof degree less than 100 such that f(1)·f(2) =0.

2. For each subspace from the previous question, compute its dimension.

3. From the previous tutorial, we know that v1 =

 0 1 1

, v2 =

 1 0 1

, and v3 =

 1 1 0

form a basis of R3. Compute the coordinates of the vector

 5 3 1

relative to this basis.

4. Show that the vectors1+t2, 2−t+t2, andt−t2 form a basis of the vector space P2 of polynomials in t of degree at most 2, and compute the coordinates of the vector t2+4t+4 relative to this basis.

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