Suppose
𝐴
is a
3
×
4
matrix and the column vectors of
𝐴
span
𝑅
3
. Which of the following statements must be FALSE?

Question

Suppose 
𝐴
 is a 
3
×
4
 matrix and the column vectors of 
𝐴
 span 
𝑅
3
. Which of the following statements must be FALSE?

BlackTom AI · Accepted Answer

We start by unpacking the given information: A is a 3×4 matrix and its column vectors span R^3. This means the column space of A has dimension 3, so rank(A) = 3. Since A has 4 columns, the number of free variables (nullity) is nullity(A) = n − rank(A) = 4 − 3 = 1. Consequently, the homogeneous system Ax = 0 has infinitely many solutions (there exists at least one nonzero solution in addition to the trivial x = 0). This already has implications for several of the statements below.

Option 1: The linear system Ax = b has a solution for all b in R^3.
- With rank(A) = 3, the column space of A is all of R^3, because the columns span R^3. Therefore, for every b in R^3, there exists at least one x such that Ax = b. This statement is consistent with the given information and is true.

Option 2: The linear system Ax = (1, 2, 3) has an infinite number of solutions.
- Since rank(A) = 3 and the number of variables is 4, the system Ax = b is either inconsistent or has infinitely many solutions depending on b, but for a b in the column space (which is all of R^3 here), there will be infinitely many solutions due to the one-dimensional nullspace. Specifically, if a particular solution x_p exists, then x = x_p + t v, where v spans the nullspace and t is any real number, gives infinitely many solutions. Thus, for b = (1,2,3), which lies in R^3, this system indeed has infinitely many solutions. This statement is true given the rank-nullity situation.

Option 3: The linear system A^T x = 0 has a unique solution.
- A^T is a 4×3 matrix. The system A^T x = 0 has as many equations as the rank can support in the transpose space. Since rank(A) = 3, the rank of A^T is also 3, and the nullspace of A^T has dimension 4 − rank(A^T) = 4 − 3 = 1. Therefore, there are infinitely many solutions to A^T x = 0 (beyond the trivial x = 0). This statement is false because the unique solution claim contradicts the actual one-dimensional nullspace of A^T.

Option 4: The linear system Ax = 0 has a unique solution.
- As established, nullity(A) = 1, so Ax = 0 has more than the trivial solution; there exists a nonzero vector in the nullspace. Hence Ax = 0 does not have a unique solution. This statement is false for the same reason that the system has a nontrivial nullspace.

MATH_2114_92069_202509 Common-Time Final

Suppose 𝐴 is a 3 × 4 matrix and the column vectors of 𝐴 span 𝑅 3 . Which of the following statements must be FALSE?

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