Unformatted text preview: Math 2090, Spring 2009. Testreview2
1. Determine all values of the constant k for which the following system has an infinite number of solutions.
2. Reduce the given matrix to upper triangular form and then evaluate the determinant.
3. Use the cofactor expansion theorem to evaluate the determinant along column 3.
4. Use the adjoint method to find A-1 if
5. Use the Cramers rule to determine x1 and x2
6. Consider the vector space and let S be the subset of V consisting of those functions satisfying the DE on I. Determine whether S is a subspace of V.
7. Let , and is he set of all vectors in V satisfying . Determine whether S is a subspace of V.
8. Determine the null space of the matrix
9. Determine whether the set is LI or LD in. In case of linear dependence, find a dependency relationship.
10. Find those x values for which the following vectors are LD
11. Determine whether the set is LI in.
SO THEY ARE L.D.
12. Show that the following functions have Wronskian
And hence determine the conditions on a, b, c such that the set is LI on any interval.
I used the transpose of A since they have the same determinant.
13. Determine whether the set is LI or LD on
14. Find those k values for which the following vectors are LD.
15. Determine if the vector lies in span where and
are in R .
16. Let S be the subspace of spanned by the vectors
. Determine a basis for S, and hence, find dim[S].
Thus Dim[S]=2 Basis is any 2 of the given vectors. 17.
In question (a) compute the Wronskian (It is not equal to zero so they are LI) In question (b)if you compute the Wronskian it will be equal to zero and by quick inspection you can see they are LD.
18. Find the dimension of the null space of the matrix
Det(A)is not equal to zero therefore Nullspace of A contains only the zero vector hence its dimension is zero.
19. Are the polynomials a basis for P3 (the vector space of polynomials of degree at most three)? Justify your answer
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