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Every elementary row operation is reversible
This statement is true
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A 5x6 matrix has 6 rows
False- it has 5 rows and 5 columns
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Two fundamental questions about a linear system involve existence and uniqueness
This statement is true
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Two matrices are row equivalent if they have the same number of rows
False- they are row equivalent if their reduced echelon form is the same; a matrix with 2 rows and 5 columns is not equivalent to a matrix with 2 rows and 2 columns
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Elementary row operations on an augments matrix never change the solution set of the associated LS
This statement is true
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Two equivalent linear systems can have different solution sets
False- In order to be equivalent, they must have the same solution set
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A consistent LS has one or more solutions
This statement is true
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In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations
False: reduced echelon form is unique so for every matrix, there is only one row reduced echelon matrix
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The row reduction algorithm applies only to augmented matrices for a LS
False- it applied to all matrices including augmented matrices and coefficient matrices
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A leading (basic) variable in a LS is a variable that corresponds to a pivot column in the coefficient matrix
This statement is true
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Finding a parametric description of the solution set of a LS is the same as solving the system
This statement is true
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If one row in an echelon form of an augmented matrix is [ 0 0 0 5 | 0], then the system is inconsistent
False- dividing the system by 5 will get [ 0 0 0 1 | 0] which will mean that X4=0 so that row does not cause the system to be inconsistent
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The reduced echelon form of a matrix is unique
This statement is true
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If every column of an augmented matrix contains a pivot, then the corresponding LS is consistent
False: if every column has a pivot, then the augmented column will also have a pivot which will cause a 0=# error leading to the system being inconsistent and having no solutions
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The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process
False: the pivot positions are based off of the LS corresponding to the matrix and on the nonzero rows of the matrix so elementary row operations wont change them
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Whenever a system has free variables, the solution set contains many solutions
False: This is only true when the LS is consistent, but if it is inconsistent, then it will have no solution regardless of the free variables
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False: [-4 over 3] is a 2x1 vector and [-4 3] is a 1x2 vector; the two vectors are transposes of each other; not equivalent
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False: they would lie on the same line if the second vector was [2, -5]
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False: the span of {u,v} could also be a line if the vectors are scalar multiples
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False: if u=[1 over 0] and v= 90 over 1], then the two vectors span the entire xy plane
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False- that results in the vector u
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False: If they are all zero then the resulting vector is the zero vector which is always in the span of a set of vectors
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False: That is the matrix equation; vector eq is x1[a1] +x2[a2]...=b
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False: If one of the pivots is in the augmented matrix then the system is inconsistent
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False: if there is a pivot position in every row, then the system may or may not be inconsistent
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False: it is inconsistent because the columns do not span Rm (theorem 4)
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False: it has a trivial solution if it has no free variables; if there are free variables, then it has a nontrivial solution
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False: It is a line through p parallel to v
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False: non trivial solution means that there is atleast 1 free variable which can be any number including zero
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False: Ax=0 must have ONLY the trivial solution
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False: to be LD, only one vector needs to be a linear combination of the others
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True because that will result in atleast 1 free variable because p>n
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False: [3,3,3] and [1,1,1]: there are two vectors with 3 entries in each vector but they are LD because they are scalar multiples
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true: LD then p>n so more than n vectors means more than n columns
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A LT is a function from Rn to Rm that assigns each vector x in Rn to a vector T(x) in Rm
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False: To multiply with A, x must be a vector with 5 entries (5 rows so R5) so the domain of T is R5
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False: A matrix transformation is a special type of linear transformation in the form x-> Ax
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False: Multiplying two linear functions will still result in a linear function
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False: that makes the vector one-to-one; it is onto if its range=codomain
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False: T can be one-to-one if both columns have a pivot position
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False: It is in the reverse order
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False: the inverse of AB is B-1 A-1
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False: ad-bc cannot be zero
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False: It is elementary row operations just not the same ones
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False: it is one to one because if the system has a trivial solution (every column is a pivot column)
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False: It could be consistent with free variables meaning that there are infinite many solutions
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