Two fundamental questions about a linear system involve existence and uniqueness
This statement is true
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
Elementary row operations on an augments matrix never change the solution set of the associated LS
This statement is true
Two equivalent linear systems can have different solution sets
False- In order to be equivalent, they must have the same solution set
A consistent LS has one or more solutions
This statement is true
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
The row reduction algorithm applies only to augmented matrices for a LS
False- it applied to all matrices including augmented matrices and coefficient matrices
A leading (basic) variable in a LS is a variable that corresponds to a pivot column in the coefficient matrix
This statement is true
Finding a parametric description of the solution set of a LS is the same as solving the system
This statement is true
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
The reduced echelon form of a matrix is unique
This statement is true
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
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
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
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
False: they would lie on the same line if the second vector was [2, -5]
This statement is true
This statement is true
False: the span of {u,v} could also be a line if the vectors are scalar multiples
False: if u=[1 over 0] and v= 90 over 1], then the two vectors span the entire xy plane
This statement is true
This statement is true
False- that results in the vector u
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
False: That is the matrix equation; vector eq is x1[a1] +x2[a2]...=b
This statement is true
False: If one of the pivots is in the augmented matrix then the system is inconsistent
This statement is true
This statement is true
This statement is true
This statement is true
This statement is true
This statement is true
False: if there is a pivot position in every row, then the system may or may not be inconsistent
This statement is true
False: it is inconsistent because the columns do not span Rm (theorem 4)
This statement is true
False: it has a trivial solution if it has no free variables; if there are free variables, then it has a nontrivial solution
False: It is a line through p parallel to v
False: non trivial solution means that there is atleast 1 free variable which can be any number including zero
This statement is true
This statement is true
This statement is true
False: Ax=0 must have ONLY the trivial solution
False: to be LD, only one vector needs to be a linear combination of the others
True because that will result in atleast 1 free variable because p>n
This statement is true
True
This statement is true
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
true: LD then p>n so more than n vectors means more than n columns
A LT is a function from Rn to Rm that assigns each vector x in Rn to a vector T(x) in Rm
False: To multiply with A, x must be a vector with 5 entries (5 rows so R5) so the domain of T is R5
False: A matrix transformation is a special type of linear transformation in the form x-> Ax
This statement is false
This statement is true
This statement is true
This statement is true
This statement is true
This statement is true
This statement is true
This statement is true
False: Multiplying two linear functions will still result in a linear function
False: that makes the vector one-to-one; it is onto if its range=codomain
False: T can be one-to-one if both columns have a pivot position
False: It maps R4 to R3
This statement is true
This statement is true
This statement is true
This statement is False
This statement is true
This statement is true
This statement is true
False: It is in the reverse order
This statement is true
False- no plus signs
False: It is Ct Bt At
This statement is true
This statement is true
False: the inverse of AB is B-1 A-1
False: ad-bc cannot be zero
This statement is true
This statement is true
False: It is elementary row operations just not the same ones
This statement is true
False: reverse order
This statement is true
This statement is true
This statement is true
This statement is true
This statement is false
This statement is true
This statement is true
This statement is true
This statement is true
This statement is true
False: it is one to one because if the system has a trivial solution (every column is a pivot column)
False: It could be consistent with free variables meaning that there are infinite many solutions
