Algebra Questions

2 -4 8 1. Let v1 = 3 and V2 = -5 Is the vector W = 2 an element of the subspace of R3 which is spanned -5 8 9 by V1 and V2 ? Justify with a supporting computation and/or explanation. 2. Suppose you are given a set S containing n vectors, denoted as = v.) Also suppose that every element of the set (kjkk.ky kn} is a real scalar. Finally, assume that the only solution of the linear system k,V,+K2V2+KVV, +k.v.= is found to be the trivial solution 0, k, = 0. Based on this information, answer the following questions concerning set S, providing a brief justification for each answer. You may cite theorem numbers from the text, if applicable, to support your justifications, but do not copy any phrasing directly from any textbook (use your own words instead to describe what is going on or why). (a) Is set S a linearly dependent set or a linearly independent set? (b) The vectors in set S span a particular vector space V. Do they also form a basis for V? (c) Is it necessarily true that the dimension of vector space V is exactly equal to n? (d) If a matrix M were formed in such a way that its column vectors consisted of the vectors from set S, would the value of the determinant det(M) be zero or non-zero? Consider the 3x4 matrix A given below: 1 1 3 1 A= 2 1 5 4 1 2 4 -1 (a) Find a basis for the column space of A. (b) Find a basis for the row space of A. (c) Find a basis for the null space of A. (d) Verify that rank(A)+nullity(A) = 4 (the number of columns of A) in accordance with theorem 4.8.2. 5. A certain vector space is spanned by 1 2 4 3 18 V2 = = 1 2 2 4 -3 -8 Because there are five vectors but only four components per vector, this set must be linearly dependent, and cannot be a basis. However, a particular subset of these vectors does form a basis; find this basis. Part (a) of Example 10 in section 4.7 of the textbook is similar. Let x be an arbitrary vector in R², and let B = {b,,b2} and C be two bases for R², where b, c, c2= (a) Determine the matrix which maps the B-coordinates of x to the C-coordinates of x (that is, which transforms [x], into [x]c) (b) Suppose that, for a particular vector x, its coordinates relative to basis 13 are [x1-Use the result of part (a) to determine [x]c.