Matrix Algebra Test 1

Suppose A is a 5×7 matrix. How many pivot columns must A have if its columns span ℝ5​? ​Why?

The matrix must have 5 pivot columns. The statements​ "A has a pivot position in every​ row" and​ "the columns of A span ℝ5​" are logically equivalent.

Suppose A is a 7×5 matrix. How many pivot columns must A have if its columns are linearly​ independent? Why?

The matrix must have 5 pivot columns.​ Otherwise, the equation Ax=0 would have a free​ variable, in which case the columns of A would be linearly dependent.

If a matrix A is 6×8 and the product AB is 6×5​, what is the size of​ B?

The size of B is 8*5

Determine whether the statement below is true or false. Justify the answer. A 5×6 matrix

The statement is false. A 5×6 matrix has five rows and six columns.

Determine whether the statement below is true or false. Justify the answer. In some​ cases, a matrix may be row reduced to more than one matrix in reduced echelon​ form, using different sequences of row operations.

The statement is false. Each matrix is row equivalent to one and only one reduced echelon matrix.

Determine whether the statement below is true or false. Justify the answer. The columns of a matrix A are linearly independent if the equation Ax=0 has the trivial solution.

The statement is false. For every matrix​ A, Ax=0 has the trivial solution. The columns of A are independent only if the equation has no solution other than the trivial solution.

Determine whether the statement below is true or false. Justify the answer. If S is a linearly dependent​ set, then each vector is a linear combination of the other vectors in S.

The statement is false. If an indexed set of​ vectors, S, is linearly​ dependent, then it is only necessary that one of the vectors is a linear combination of the other vectors in the set.

Determine whether the statement below is true or false. Justify the answer. The set Span {u, v} is always visualized as a plane through the origin.

The statement is false. It is often​ true, but Span {u, v} is not a plane when v is a multiple of u or when u is the zero vector.

Determine whether the statement below is true or false. Justify the answer. The echelon form of a matrix is unique.

The statement is false. The echelon form of a matrix is not​ unique, but the reduced echelon form is unique.

Determine whether the statement below is true or false. Justify the answer. The solution set of a linear system involving variables x1​, ​..., xn is a list of numbers s1, ..., sn that makes each equation in the system a true statement when the values s1​, ​..., sn are substituted for x1​, ​..., xn​, respectively.

The statement is false. The given description is of a single solution of such a system. The solution set of the system consists of all possible solutions.

Determine whether the statement below is true or false. Justify the answer. The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process.

The statement is false. The pivot positions in a matrix are determined completely by the positions of the leading entries in the nonzero rows of any echelon form obtained from the matrix.

Determine whether the statement below is true or false. Justify the answer. The solution set of Ax=b is the set of all vectors of the form w=p+vh​, where vh is any solution of the equation Ax=0.

The statement is false. The solution set could be empty. The statement is only true when the equation Ax=b is consistent for some given b​, and there exists a vector p such that p is a solution.

Question content area top Part 1 Determine whether the statement below is true or false. Justify the answer. Two fundamental questions about a linear system involve existence and uniqueness.

The statement is false. The two fundamental questions are about whether it is possible to solve the system with row operations or whether a computer is necessary.

Determine whether the statement below is true or false. Justify the answer. Two matrices are row equivalent if they have the same number of rows.

The statement is false. Two matrices are row equivalent if there exists a sequence of elementary row operations that transforms one matrix into the other.

Determine whether the statement below is true or false. Justify the answer. The columns of any 4×5 matrix are linearly dependent.

The statement is true. A 4×5 matrix has more columns than​ rows, and if a set contains more vectors than there are entries in each​ vector, then the set is linearly dependent.

Determine whether the statement below is true or false. Justify the answer. The equation Ax=b is homogeneous if the zero vector is a solution.

The statement is true. A system of linear equations is said to be homogeneous if it can be written in the form Ax=0​, where A is an m×n matrix and 0 is the zero vector in ℝm. If the zero vector is a​ solution, then b=Ax=A0=0.

Determine whether the statement below is true or false. Justify the answer. Elementary row operations on an augmented matrix never change the solution set of the associated linear system.

The statement is true. Each elementary row operation replaces a system with an equivalent system.

Determine whether the statement below is true or false. Justify the answer. Every elementary row operation is reversible.

The statement is true.​ Replacement, interchanging, and scaling are all reversible.

Determine whether the statement below is true or false. Justify the answer. The equation Ax=b is consistent if the augmented matrix Ab has a pivot position in every row.

This statement is false. If the augmented matrix Ab has a pivot position in every​ row, the equation Ax=b may or may not be consistent. One pivot position may be in the column representing b.