8. DETAILS LARLINALGEM 4.R.084. MY NOTES ASK YOUR TEACHER Determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) The set W = {(0,x2, x3): x2 and x3 are real numbers) is a subspace of R3. O False, this set is not closed under addition or scalar multiplication. O False, this set is closed under addition but is not closed under scalar multiplication. True, this is the standard subspace of R3. O False, this set is not closed under addition but is closed under scalar multiplication. True, this set is closed under addition and scalar multiplication. (b) A set of vectors S in a vector space V is a basis for V when S spans V and S is linearly independent. O False, V is a basis for S. O False, V needs to span S for it to be a basis. False, S needs to span V and S needs to be linearly dependent. O True, this is the definition of a basis. O False, we need the dimensions of S and V to guarantee a basis. (c) If A is an invertible nxn matrix, then the n row vectors of A are linearly dependent. True, this is true for nxn matrices in general. True, this follows from being invertible. False, the nxn identity matrix is invertible, but the row vectors are linearly independent. O False, this is only true for matrices where the number of columns is greater than the number of rows. O False, this is only true for matrices where the number of rows is greater than the number of columns. Submit