Which of the following statements are correct? For an nxn matrix A, the columns

Question

Which of the following statements are correct? For an nxn matrix A, the columns of A are linearly independent if and only if the rows of A are linearly independent. For an mxn matrix A, the nullity of A equals the nullity of its transpose AT. An mxn matrix A defines some linear transformation TA: Rn rightarrow Rm. TA is onto if and only if rank A = m. A set S of vectors forms a basis for a subspace V of Rn if and only if the vectors of S are linearly independent and the number of vectors in S equals the dimension of V. The set v = {[x1 x2 x3] epsilon R3:3x1 + 2x2 - x3 = 1} is not a subspace of R3.

Explanation / Answer

(a)True. This is also equivalent to saying that A isinvertible. (b)False. It would be true if m=n; in fact N(A') =m-n+N(A). (c)True. Also note that rank A =n iff TA is1-1. Here TA(x)=Ax. (d)True. (e)True. Clearly (0,0,0) is not in the subspace (as 3.0+2.0-0is not 1) which is necessary for a set to be a subspace.

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