Answers to Quizlet 4.5
- If there exists a set of p vectors which span a vector space V, then the
dimension of V is less than or equal to p.
Answer: True, as the statement is equivalent to saying that a subset
of these p vectors would form a basis for V.
- Recall that every polynomial can be thought of as a vector, a vector determined
by the coefficiants of the polynomial. How many polynomials does it take to
span the space of all quadratic (degree 2) polynomials in a single variable?
Answer: 3. Any quadratic polynomial in the single variable x can be
written in the form a x^2 +b x +c. That's three different coefficients so this
space of polynomials is of dimension 3, requiring three basis vectors (i.e.
three polynomials).
- If p=2 or p>2 and dim V=p then every set of p-1 nonzero vectors is linearly
independent.
Answer: False, consider R^3 and the set of vectors (1,0,0) and (2,0,0).
Thats (3-1) vectors and they are linearly dependent.
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