Mark each statement True or False Justify each answer Here A

Mark each statement True or False. Justify each answer. Here A is an mxn matrix. Complete parts (a) through (e) below. If 8 is a basis for a subspace H. then each vector in H can be written in only one way as a linear combination of the vectors in 8. Choose the correct answer below. The statement is true. Suppose 8 ={v_1,...,v_p} and x is a vector in H that can be generated two ways. Say. x = c_1v_1 +... + c_pv_p and x = +... +d_pv_p. then 0 = x- x= (c_1 -d_1)v_1 +... + (c_p -d_p)v_p. Therefore. c_p = d_p and x can only be generated in one way. The statement is false. Bases for a subspace H may be linear dependent and therefore there can be multiple solutions for the same vector x in H. The statement is true. All bases for a subspace H are linearly independent and therefore each vector in H can only be generated as one unique linear combination of the vectors in 8. The statement is false. Suppose 8 ={v_1,....v_p} and x is a vector in H. The vector x can be generated in a multiple of ways based on the values of the vectors in the set 8 = {v_1...v_p}. If 8 ={v_1,....v_p} is a basis for a subspace H of R^n. then the correspondence x^[x]6 makes H look and act the same as R_p. Choose the correct answer below. The statement is true. The correspondence of xiRight arrow[x]_B implies a one-to-one correspondence between H and R^p that preserves linear combinations.. The statement is false. The vectors in H may contain more than p entries and therefore the correspondence xi >[x]6 does not make H look and act the same as R_p The dimension of Nul A is the number of variables in the equation Ax = 0. Choose the correct answer below.. The statement is true. The number of total variables involved in solving the equation Ax = 0 is the dimension of Nul A. The statement is false. The dimension of Nul A is the number of free variables in the equation Ax = 0. The statement is true. The dimension of Nul A is the same as the amount of vectors in the set x = {x_1,....x_p} that satisfy the equation Ax = 0. The statement is false. The dimension of Nul A is the number of variables in the equation Ax = 0 minus the number of free variables in the equation Ax = 0. d. The dimension of the column space of A is rank A. Choose the correct answer below. The statement is true. The rank of a matrix A is the dimension of the column space of A. The statement is false. The rank of a matrix A is equal to the dimension of the column space of A plus the dimension of the null space of A. The statement is true. The rank of an m xn matrix A is equal to n, which is also equal to the dimension of the column space of A. The statement is false. The rank of an m x n matrix A is equal to the number f columns + the number of rows of A. or rank A = m + n. If H is a p-dimensional subspace of H. then a linearly independent set of p vectors in H is a basis for H. Choose the correct answer below. The statement is true. Any set of p linearly independent vectors is a basis for H. The statement is false. This is only true if n = p. The statement is false. It is possible for p vectors to be linearly independent without spanning H. The statement is false. This is only true if n not equel to p. O The statement is true. The fact that 8 = {v_1-. v_p} implies that the correspondence xi >[x]6 makes H look and act the same as IRP. O The statement is false. The correspondence of x> >[x]5 does not imply a one-to-one correspondence between H and IRP that preserves linear combinations.

Solution

c)The dimension of Nul A is the number of variables in the equation Ax=0

FALSE It’s the number of free variables

So the answer is B.

Please send the others questions in differents post