Show that a set of vectors is a basis for a vector space. Find the coordinates o

Question

Show that a set of vectors is a basis for a vector space. Find the coordinates of a vector relative to a basis. In parts (a) - (e) determine whether the statement is true or false, and justify your answer. (a) If V = span [v_1...v_n], then [v_1...v_n] is a basis for V. (b) Every linearly independent subset of a vector space V is a basis for V. Find the coordinate vector of a vector relative to a basis. (c) If [v_1...v_2...v_n] is a basis for a vector space V, then every vector in V can be expressed as a linear combination of v_1, v_2, .... v_n. (d) The coordinate vector of a vector x in R^n relative to the standard basis for R^n is x. (e) Every basis of P_4 contains at least one polynomial of degree 3 or less.

Explanation / Answer

(a). The statement is false. The basis vectors have to be linearly independent also which has not been stated here. Spanning alone is not an adequate criterion for a basis.

(b). The statement is false.The vectors in the basis must also span the vector space. As a matter of fact, every proper subset of the basis is linearly independent, but this subset does not span the vector space.

(c ).The statement is true. Since the set {v1,v2,…vn} spans V, every vector in Vcan be expressed as a linear combination of v1,v2,…vn.

(d).The statement is true. The coordinate vector of a vector x in Rn, with respect to the standard basis will be x itself as the standard basis is {e1,e2,…en}, where ei,(1in) has 1 in the ith position and a 0 in all other positions.

(e ). The statement is true.An arbitrary element of P4 is ax4+bx3+cx2+dx+e, where a,b,c,d and e are arbitrary real numbers. Hence, in every basis of P4, there must be atleast one polynomial of degree 3 or less.

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