Question

Determine whether each of statements a through f below are true or false. Justify each answer.

a. Every matrix equation Ax b corresponds to a vector equation with the same solution set. Choose the correct answer below.

A. True. The matrix equation Ax-bis simply another notation for the vector equation x1a1 +x2a2 +···+xnan-b, where a 1
B. False. The matrix equation Ax b only corresponds to an inconsistent system of vector equations.
C. False The matrix equation Ax = b does not correspond to a vector equation with the same solution set
D. True. The matrix equation Ax-b is simply another notation for the vector equation x1a1 + x2a2 +···+xnan-b, where al , , an are the columns of A , an are the rows of A.

b. If the equation Ax = b is consistent, then b is in the set spanned by the columns of A. Choose the correct answer below.

A. False. Ax= b is only consistent if the values of b are nonzero

B.False, b is only included in the set spanned by the columns of A if Ax= b is inconsistent.

C.True. The equation Ax = b has a nonempty solution set if and only if b is a linear combination of the columns of A.

D. True. The equation Ax= b has a solution set if and only if A has a pivot position in every row.

c. Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x. Choose the correct answer below.

A. True. The matrix A is the matrix of coefficients of the system of vectors.

B. False. A and x can only be written as a linear combination of vectors if and only if in Ax= b, b is nonzero.

C. False. A and x cannot be written as a linear combination because the matrices do not have the same dimensions. Click to select your answer.

Answer 1

Answer: The options are not easy to read, I'll explain the answer to a question even if I can't choose from the options.

Step-by-step explanation:

a. Every matrix equation Ax = b corresponds to a vector equation with the same solution set.

TRUE.

a. Let the vector equation of a linear system be

a1x1 + a2x2 + a3x3 + ... + anxn = b

The solution of the above vector equation is the same as the solution set of Ax = b, where A = [a1 a2 a3 ... an]. Where a1, a2, a3, ... , an, b are all column vectors.

b. If the equation Ax = b is consistent, then b is in the set spanned by the columns of A.

Option C, TRUE

Suppose A is an m×n matrix, the equation

Ax = b

is consistent if and only if the columns of A span R^m, that is every b in R^m is in the span of the columns of A, which is another way of saying that any b is a linear combination of the columns.

c. Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x.

Option A, TRUE

For a suitable matrix A and vector x, Ax can be written as a linear combination of the columns because the matrix A is the matrix of coefficients of the system of vectors, and nothing changes.