Linear Algrbra T/F problem, please do all the questions. I will definitetly give

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Linear Algrbra T/F problem, please do all the questions. I will definitetly give a thumb up thank you!

3. Mark each statement True or False. Justify each answer i. A subset H of R" is a subspace if the zero vector is in H. ii. If B is an echelon form of a matrix A, then the pivot columns of B form a basis for ColA. ii. Given vectors vi,... ,Vp in R", the set of all linear combinations of these vectors is a subspace of IR" iv. Let H be a subspace of R". If x is in H, and y is in R", then x +y is in H. V. The column space of a matrix A is the set of solutions of Ax = b.

Explanation / Answer

3. i. False. If H is a subset of R2 defined by H = {( x,y)T| xy = 0}, then the zero vector (0,0)T H, the vectors (1,0)T and (0,1)T also H, but the vector (1,0)T+(0,1)T = (1,1)T H. Hence H is not a vector space and, therefore, not a subspace of R2.

ii. False. We must scrutinize the corresponding columns of the matrix A.

iii.   False. The zero vector also must be in the set.

iv. False. y also must be in H if x+y is to be in H.

v. False. If A = I2, then col(A) = R2 . However, the vector (1,1)T col(A) is not a solution to the equation Ax = (1,0)T.

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