Question

DETAILS LARLINALG8 4.R.084. 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. () The set w = {(0,x2,x): and X" are real numbers) is a subspace of R. False, this set is not closed under addition but is closed under scalar multiplication. True, this is the standard subspace of R. false, this set is closed under addition but is not closed under scalar multiplication. Falso, this set is not closed under addition or scalar multiplication. True, this set is closed under addition and scalar multiplication, (b) A set of vectors in a vector space Vis a basis for when s spans V and S is linearly independent. false, V needs to span S for it to be a basis. False, S needs to span V and needs to be linearly dependent. True, this is the definition of a basis. False, we need the dimensions of S and V to guarantee a basis. False, V is a basis for s. (C) If A is an invertible n x n matrix, then the n row vectors of A are linearly dependent False, then x n identity matrix is invertible, but the row vectors are linearly independent True, this follows from being invertible. False, this is only true for matrices where the number of rows is greater than the number of columns. False, this is only true for matrices where the number of columns is greater than the number of rows. True, this is true for nxn matrices in general.

Answer 1

1)false , this is not closed under addition or scaler multiplication .

as if we take x2+x3 , where x2 and x3 are reals so , their sum does not contained in W and for any scaler a , a.x2 does not belong to W so , not closed under both operation .

2)true, this is defination of basis.

3)false , the n * n identirty matrix is invertible , but the row vectors are linaerly independent .

when matrix is invertible it is row equivalent to identity matrix , so like identily matrix of order n*n , our matrix also have n pivots hence , rows are said to be linearly independent .