Linear Algebra Midterm 1

Every elementary row operation is reversible. True/False?

True, because replacement, interchanging, and scaling are all reversible.

A 5×6 matrix has six rows. True/False?

False because 5×6 matrix has five rows and six columns. (Row by Columns)

Is the statement "The solution set of 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" True/False?

False, because the description applies to a single solution. The solution set consists of all possible solutions.

Is the statement "Two fundamental questions about a linear system involve existence and uniqueness" True/False?

True, because two fundamental questions address whether the solution exists and whether there is only one solution.

Two matrices are row equivalent if they have the same number of rows. True/False?

False, because if two matrices are row equivalent it means that there exists a sequence of row operations that transforms one matrix to the other.

Elementary row operations on an augmented matrix never change the solution set of the associated linear system. True/False?

True, because the elementary row operations replace a system with an equivalent system.

Two equivalent linear system can have different solution sets. True/False?

False, because two systems are called equivalent if they have the same solution set.

A consistent system of linear equations has one or more solutions. True/False?

True, a consistent system is defined as a system that has at least one solution.

In some cases a matrix may be row reduced to more than one matrix in reduced echelon from, using different sequences of row operations. True/False?

False, each matrix is row equivalent to one and only one reduced echelon matrix.

The row reduction algorithm applies only to augmented matrices for a linear system. True/False?

False, algorithm applies to any matrix, whether or not the matrix is viewed as an augmented matrix for a linear system.

A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix.

True, it is the definition of a basic variable.

Finding a parametric description of the solution set of a linear system is the same as solving the system. True/False?

False, the solution set of a linear system can only be expressed using a parametric description if the system has at least one solution.

If one row in an echelon form of an augmented matrix is [0 0 0 5 0], then the associated linear system is inconsistent.

False, the indicated row corresponds to the equation 5x₄=0, which does not by itself make the system inconsistent.

What must be true for a linear system for it to have a unique solution?

The system has no free variables, and is consistent.

Linear system is consistent when

rightmost column of the augmented matrix is not a pivot column

Suppose the coefficient matrix of a system of linear equations has a pivot position in every row. Explain why the system is consistent.

The system is consistent because the rightmost column of the augmented matrix is not a pivot column.

Suppose a 6×8 coefficient matrix for a system has six pivot columns. Is the system consistent?

Yes, because there is a pivot position in each row of the coefficient matrix. The augmented matrix will have nine columns and will not have a row of the form [0 0 0 0 0 0 0 0 1].

A system of linear equations with fewer equations than unknowns is sometimes called underdetermined system. Can such a system have a unique solution?

No, it cannot have a unique solution. Bc there are more variables than equations, there must be at least one free variable. If the linear system is consistent there is at least one free variable, the solution set contains infinitely many solutions. If the linear system is inconsistent, there is no solution.

When u and v are nonzero vectors, Span {u,v} contains only the line through vector u and the line through vector v and the origin. True/False?

False, Span{u,v} includes linear combinations of both u and v vectors.

Any list of five real numbers is a vector in R^5. True/False?

True, R^5 denotes the collection of all lists of five real numbers.

Asking whether the linear system corresponding to an augmented matrix [a₁ a₂ a₃ b] has solution amounts to asking whether b is in span {a₁, a₂, a₃}. True/False?

True, an augmented matrix has a solution when the last column can be written as a linear combination of the other columns. A linear system augmented has a solution when the last column of its augmented matrix can be written as a linear combination of the other columns.

The vector v results when a vector u-v is added to the vector v. True/False?

False, adding u-v to v results in u.

The weights c₁,…,c(subp) in a linear combination c₁v₁+…+c(subp)v(subp) cannot all be zero. True/False?

False, setting all the weights equal to zero results in the vector 0.

What doe sit mean when the row echelon form of the augmented matrix [A b] demonstrates that there is no pivot position in the third row of A/every row?

Therefore, Ax=b does not have a solution for all possible b. Ax=b has a solution only when all rows of the augmented matrix [A b] represent true equations not [0 0 0 b]

The equation Ax=b is referred to as a vector equation. Choose the correct answer below. True/False?

False, the equation Ax=b is referred to as a matrix equation because A is a matrix.

A vector b is a linear combination of the columns of a matrix A if and only if the equation Ax=b has at least one solution. True/False?

True, the equation Ax=b has the same solutions set as the equation x₁a₁+x₂a₂+…+x(subn)a(subn)=b

The equation Ax=b is consistent if the augmented matrix [A b] has a pivot position in every row.

False, is the augmented matrix [A b] 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.

The first entry in the product Ax is a sum of products. True/False?

True, the first entry in Ax is the sum of products of corresponding entries in x and the first entry in each column of A.

If the columns of mxn matrix A span R^m, then the equation Ax=b is consistent for each b in R^m.

True, if the columns of A span R^m, then the equation Ax=b has a solution for each b in R^m.

If A is an mxn matrix and if the equation Ax=b is inconsistent for some b in R^m then A cannot have a pivot position in every row.

True, if A is an mxn matrix and if the equation Ax=b is inconsistent for some b in R^m, then the equation Ax=b has no solution for some b in R^m

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

True, the matrix equation Ax=b is simply another notation for the vector equation x₁a₁+x₂a₂+…+x(subn)a(subn)=b, where a₁,…,a(subn) are the columns of A.

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

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

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

True, the matrix A is the matrix of coefficient of the system of vectors.

If the coefficient matrix A has a pivot position in every row, then the equation Ax=b is inconsistent.

False, if A has a pivot position in every row, the echelon form of the augmented matrix could not have a row such as [0 0 0 1] and Ax=b must be consistent.

system has a nontrivial solution if there is a free variable after RREF. True/False? Is it linearly dependent or independent?

True, free variables makes it nontrivial solution. Linearly dependent

Linearly independent means it has a trivial solution

True