Page 49 - Linear Algebra Done Right

Exercises
Exercises
1. Prove that if (v 1 ,...,v n ) spans V, then so does the list 35
(v 1 − v 2 ,v 2 − v 3 ,...,v n−1 − v n ,v n )
obtained by subtracting from each vector (except the last one)
the following vector.

2. Prove that if (v 1 ,...,v n ) is linearly independent in V, then so is
the list
(v 1 − v 2 ,v 2 − v 3 ,...,v n−1 − v n ,v n )
obtained by subtracting from each vector (except the last one)
the following vector.
3. Suppose (v 1 ,...,v n ) is linearly independent in V and w ∈ V.
Prove that if (v 1 + w,...,v n + w) is linearly dependent, then
w ∈ span(v 1 ,...,v n ).
4. Suppose m is a positive integer. Is the set consisting of 0 and all
polynomials with coefficients in F and with degree equal to m a
subspace of P(F)?
5. Prove that F ∞ is infinite dimensional.
6. Prove that the real vector space consisting of all continuous real-
valued functions on the interval [0, 1] is infinite dimensional.
7. Prove that V is infinite dimensional if and only if there is a se-
quence v 1 ,v 2 ,... of vectors in V such that (v 1 ,...,v n ) is linearly
independent for every positive integer n.
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8. Let U be the subspace of R defined by
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U ={(x 1 ,x 2 ,x 3 ,x 4 ,x 5 ) ∈ R : x 1 = 3x 2 and x 3 = 7x 4 }.
Find a basis of U.

9. Prove or disprove: there exists a basis (p 0 ,p 1 ,p 2 ,p 3 ) of P 3 (F)
such that none of the polynomials p 0 ,p 1 ,p 2 ,p 3 has degree 2.

10. Suppose that V is finite dimensional, with dim V = n. Prove that
there exist one-dimensional subspaces U 1 ,...,U n of V such that
V = U 1 ⊕· · · U n .