2. (15 points) For each of the following, determine if the statement is true or

Question

2. (15 points) For each of the following, determine if the statement is true or false. If the statement is true you must clearly explain why. If the statement is false you must clearly explain why, or give a counterex- ample. a) If a set of vectors in Rn contains the zero vector, then the set is linearly dependent (b) If u and v are linearly independent and w is in span [u, v then u, v, w is linearly dependent. (c) The columns of a 3 × 5 matrix are linearly dependent (d) The columns of a 5 3 matrix are linearly independent

Explanation / Answer

a) is True

“if a set of vectors in Rn contains the zero vector then the set is linearly dependent “ statement is true where Rn = {v1,v2,v3 ……,vn} are linearly independent. c1v1+c2v2+c3v3 +……+cnvn =0 iff c=cn=0.So by definition, any set of vectors that contain the zero vectors is linearly dependent.

b) is True

Take R ={u,v,w} is a linearly dependent set with the linearly independent subset S={u,v}.

c) is True

we have 5 column vectors of R3 (3 rows). Thus they can never be linearly independent, because the dimension of R3 is 3.The columns of a 3X5 matrix  are linearly dependent iff rank(matix)=5.

d) is True

we have 3 column vectors of R5 (5 rows). Thus they can be linearly independent, because the dimension of R5 is 5.The columns of a 5X3 matrix  are linearly independent iff rank(matix)=3. Where rank(matrix)min{m,n} that the columns of an m×n matrix  are linearly independent if m>=n.

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