Linear Final
A consistent system of four linear equations in three variable corresponds to an augmented matrix whose RREF has two pivots. Complete the following statements: 1) The solution set for the system is a _____________ 2) Each solution to the system of linear equations is in _________
1) a line 2) R^3
Suppose A is an 8 × 5 matrix, and the range of the transformation T(x) = Ax is a line. Fill in the blank: Nul(A) is a ______-dimensional subspace of R^___
4,5 dim(Nul A) = 5 − dim(ColA) = 5 − 1 = 4, and Nul A is a subspace of R^5
Suppose that A is a 5×7 matrix that has an echelon form with no zero rows. Find the dimension of the column space of A, and the dimension of the null space of A .The dimension of the column space of A is _____ The dimension of the null space of A is____
5, 2
Let A be a 8 x 5 matrix. What must m and n be if we define the transformation T: R^m --> R^n by T(x)=Ax m=___ n=____
5, 8
Let A be a matrix with linearly independent columns. Which of these statements must be true? A. The equation Ax=b has a solution for all b precisely when it is a square matrix. B. There is no easy way to tell if Ax=b has a solution for all b. C. The equation Ax=b never has a solution for all b. D. The equation Ax=b always has a solution for all b. E. The equation Ax=b has a solution for all b precisely when it has more rows than columns. F. The equation Ax=b has a solution for all b precisely when it has more columns than rows. G. none of the above
A
inhomogeneous
A system of linear equations of the form Ax=b with b not = 0
Which are true? A. Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x B. The equation Ax=b is consistent if the augmented matrix [A∣b] has a pivot position in every row C. The equation Ax=b is referred to as a vector equation D. If the equation Ax=b is inconsistent, then b is not in the set spanned by the columns of A. E. Every matrix equation Ax=b corresponds to a vector equation with the same solution set. F. 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
A., D., E.
Let A be a matrix with more rows than columns. Which one of the following statements must be true? A. The columns of A are linearly independent, as long as they do not include the zero vector. B. The columns of A could be either linearly dependent or linearly independent. C. The columns of A must be linearly independent. D. The columns of A must be linearly dependent. E. The columns of A are linearly independent, as long as no column is a scalar multiple of another column in A F. none of the above
B
Let V be the subset of R^3 consisting of the vectors [a b c] with abc=0 True or False? a) V contains the zero vector b) V is closed under vector addition, meaning that if u and v are in V then u+v is in V c)V is closed under scalar multiplication, meaning that if u is in V and c is a real number then cu is in V d) V is a subspace of R^3
a) true b) false c) true d) false
Write a matrix A with the property that the equation Ax=(1 1 0) is consistent
any matric with (1 1 0) in one of its columns and the rest zeros
A system of 3 linear equations in 4 variables can have exactly one solution
false
For any matrices A and B, if the product AB is defined, then BA is also defined
false
If A is a 4 x 5 matrix and the solution set to Ax=0 is a line, then Ax=b must be inconsistent for some b in R^4
false
If A is a 5×4 matrix, and B is a 4×3 matrix, then the entry of AB in the 3rd row / 2nd column is obtained by multiplying the 3rd column of A by the 2nd row of B
false
If A is an m × n matrix with more columns than rows, then Ax = b must be inconsistent for some b in R^m
false
If A is an m×n matrix and m > n, then the matrix transformation T(x) = Ax cannot be one-to-one
false
If B is an echelon form of a matrix A, then the pivot columns of B form a basis of the column space of A
false
If S is a set of linearly dependent vectors, then every vector in S can be written as a linear combination of the other vectors in S
false
If a linear system has four equations and seven variables, then it must have infinitely many solutions.
false
If a set S of vectors contains fewer vectors than there are entries in the vectors, then the set must be linearly independent.
false
If an augmented matrix in reduced row echelon form has 2 rows and 3 columns (to the left of the vertical bar), then the corresponding linear system has infinitely many solutions.
false
If the number of rows of an augmented matrix in reduced row echelon form is greater than the number of columns (to the left of the vertical bar), then the corresponding linear system has infinitely many solutions.
false
If x is a nontrivial solution of Ax=0, then every entry of x is nonzero
false
Suppose T : R 4 → R 3 is a transformation. Then for each y in R 3 , there is a vector x in R 4 so that T(x) = y
false
Suppose T : R^4 → R^2 and U : R^2 → R^3 are matrix transformations, and let A be the standard matrix for U ◦ T, so (U ◦ T)(x) = Ax. Then A is a 4 × 3 matrix
false
Suppose u, v, and w are vectors in R^3 . Then Span{u, v, w} is either a plane in R^3 or all of R^3
false
Suppose v1 , v2 , and v3 are linearly dependent vectors in R 4 . Then v1 must be a linear combination of v2 and v3
false
Suppose v1 , v2 , v3 , v4 are vectors in R 5 , so that Span{v1 , v2 } has dimension 2 and Span{v3 , v4 } has dimension 2. Then Span{v1 , v2 , v3 , v4 } has dimension 4
false
Suppose we are given a consistent system of 1 linear equation in 3 variables and the corresponding augmented matrix has 1 pivot in its reduced row echelon form. Then the set of solutions to the equation must be a line
false
The columns of a matrix A are linearly independent if the equation Ax=0 has the trivial solution
false
The homogeneous system Ax=0 as the trivial solution if and only if the system has at least one free variable
false
The null space of an m x n matrix is a subspace of R^m
false
The plane 2x+y-z=3 contains the point (1,1,1)
false
The set of all solutions (x, y, z) to the equation x-y-z-0 is a line in R^3
false
There is a 3×4 matrix whose null space is a plane and whose column space is a line
false
There is a 4 × 7 matrix A that satisfies dim(NulA) = 1
false
There is a vector [b1 b2] so that the set of solutions (1 0 1 0 1 0 ) [x1 x2 x2]= [b1 b2] is the z-axis
false
There is an n × n matrix A so that the zero vector is an eigenvector of A.
false
The vector equation x1(1 -1) + x2(2 -2) = (2 2) is consistent
false ( put it in RREF and there is a pivot in rightmost column)
Suppose A is a 3 x 4 matrix. Which of the following are true? i) The homogeneous equation Ax=0 must have infinitely many solutions ii) If v is in the column span of A, then v is in R^4 iii) If A has a pivot in its rightmost column, then the equation Ax=0 is inconsistent iv) The equation Ax=b must be inconsistent for some vector b in R^3
i)
Suppose {v1,v2,v3} is a set of vectors in R^n. Which of the following statements are true? i) If {v1,v2,v3} is a basis for R^n ,then n=3 ii)If the vector equation x1 v1 + x2 v2 + x3 v3 = 0 has the trivial solution, then {v1 , v2 , v3 } must be linearly independent iii) If {v1 , v2 , v3 } is linearly dependent, then there is a nonzero number x1 , a nonzero number x2 , and a nonzero number x3 so that x1 v1 + x2 v2 + x3 v3 = 0.
i)
Suppose {v1 , v2 , v3 , v4 } is a linearly independent set of vectors in R 4 . Which of the following statements are true? i) For each b in R 4 , the vector equation x1 v1 + x2 v2 + x3 v3 + x4 v4 = b is consistent and has a unique solution ii)It is possible that the set {v1 , v2 , v3 } is linearly dependent. iii) Span{v1 , v2 , v3 , v4 } = R 4 .
i) iii)
Suppose that A is a matrix that represents a linear transformation T from R 7 to R 9 . In other words, T is the transformation given by the formula T(x) = Ax i) How many rows does A have? ii) Suppose the reduced row echelon form of the matrix A contains 3 pivots. Apply the Rank Theorem to A to fill in the following blanks with numbers : dim (Col A)=_________ dim (Nul A)= _________
i) 9 ii) 3, 4
Suppose A is a 2x3 matric and v is some vector so that the set of solutions to Ax=v has parametric form x1=1+x3 x2=2-x3 x3=x3 Which are true? i) The solution set for Ax=0 is span {(1 -1 1)} ii)For each b in R^2, the equation Ax=b is consistent iii) v is not the zero vector
i) ii) iii)
Suppose {u, v, w} is a basis for some subspace V of R n . Which of the following must be true? i) If {a, b,c} are vectors in V and Span{a, b,c} = V, then {a, b,c} must be a basis for V ii) The set {u, u + 2v, v + w} must be a basis for V iii) If {a, b,c} is any set of 3 linearly independent vectors in V, then {a, b,c} must be a basis for V
i) ii) iii)
consider the set V= {(x y) in R^2 | x-y ≥ 0)} i) Does V contain the zero vector? ii) Is V closed under addition? In other words, if u and v are in V, must it be true that u + v is in V? iii) Is V closed under scalar multiplication? In other words, if c is a real number and u is in V, must it be true that cu is in V?
i) yes ii) yes iii) no
Suppose A is an n×n matrix. Which of the following conditions guarantee that λ = 4 is an eigenvalue of A? i)The equation (A− 4I)x = 0 has infinitely many solutions ii)There is a nonzero vector x in R n so that the set {x,Ax} is linearly dependent iii)There is a non-trivial solution to the equation Ax = 4x iv)Nul(A− 4I) = {0}
i), iii)
Which of the following transformations are linear transformations? Clearly circle all that apply. (i) T : R 3 → R 3 given by T(x1 , x2 , x3 ) = (x1 − x2 , 1 − x1 , x1 ). (ii) T : R 2 → R 3 given by T(x1 , x2 ) = (0, x1 , x1 ). (iii) T : R 2 → R 2 given by T(x1 , x2 ) = (x1 , x1 x2 )
ii)
Let A and B be 3 × 3 matrices satisfying det(A) = 2 and det(B) = −3. Which of the following must be true? i) det(A+ B) = det(A) + det(B) ii)det(A^T B^−1 ) = −2/3 iii)det(−2A) = −16
ii) , iii)
Suppose v1, v2, and b are vectors in R^3. Which of the following are true? i) The vector equation x1v1+x2v2=b corresponds to a system of two linear equations in three variables ii) If the vector equation x1v1+x2v2=b has a solution, then some vector w in R^3 is not in Span{v1,v2,b} iii) If x=(0 0) is a solution to the equation x1v1+x2v2=b, then b= (0 0 0)
ii) and iii)
Suppose A is a 3×4 matrix and B is a 4×5 matrix, and let T be the matrix transformation T(x) = ABx. Which of the following must be true? i) The null space of AB is a subspace of R 4 ii) Every vector in the column space of AB is also in the column space of A iii) T cannot be one-to-one.
ii) iii)
Which of the following transformations are linear? (i) T : R 2 → R 2 given by T(x1 , x2 ) = (x1 ,|x2 |). (ii) T : R 3 → R 3 given by T(x1 , x2 , x3 ) = (x1 − x2 , x3 , x1 ). (iii) T : R 2 → R 2 that reflects vectors across the line y = −x
ii) iii)
Let v1=(1 -2 1) and v2= (2 -4 2) Which of the following are true? i) Span {v1,v2} is a plabe in R^3 ii) The vector w= (1 -2 0) is a linear combination of v1 and v2 iii) If b is a vector and the vector equation x1v1+x2v2=b is consistent then the solution set is a line in R^2
iii)
Suppose that T : R n → R m is a linear transformation with standard matrix A. Which of the following conditions guarantee that T is one-to-one? (i) For each x in R^n , there is a unique y in R^m so that T(x) = y. (ii) For each y in R^m, the matrix equation Ax = y is consistent. (iii) The columns of A are linearly independent
iii)
Suppose we are given a system of 3 linear equations in 3 variables. Which of the following statements are true? i)If the system is consistent, then it must have exactly one solution. ii)The system must be consistent if its corresponding augmented matrix has 3 pivots iii)The system must be consistent if the RREF of the corresponding augmented matrix has bottom row equal to ( 0 0 1 | -2) iv) One solution to the system must be the trivial solution
iii)
Suppose A is a 4×3 matrix and B is a 3×2 matrix, and let T be the matrix transformation T(x) = ABx. Which of the following must be true? (i) The column space of AB is a subspace of R 2 . (ii) Every vector in the null space of AB is also in the null space of A. (iii) T has domain R 2 and codomain R 4 . (iv) T cannot be onto
iii) iv)
Suppose A is an 11 × 5 matrix and T is the corresponding linear transformation given by the formula T(x) = Ax. Which of the following statements are true? i) dim(Col A) ≥ dim(NulA). ii) If the columns of A are linearly independent, then the range of T is R^5 iii) Suppose b is a vector so that the matrix equation Ax = b is consistent. Then the set of solutions to Ax = b must be a subspace of R^5 iv)If the matrix equation Ax = 0 has infinitely many solutions, then rank(A) ≤ 4
iv)
homogeneous
linear equations where everything to the right of the = is zero
T is a one-to one transformation from R^n to R^m. What can one say about the relationship between m and n?
n less than or equal to m
T a transformation from R^n to R^m if n>m is it one-to-one?
no
Suppose A is a 3 × 3 matrix and b1 and b2 are vectors in R^3 . Answer each of the following questions: Is it possible for (A | b1) to have a unique solution and (A | b2) to have infinitely many solutions? Is it possible for (A | b1) too have a unique solution and (A | b2) to be inconsistent?
no no
Suppose we are given a consistent linear system of 4 equations in 5 variables, and suppose that the augmented matrix corresponding to the system has 3 pivots. Then the solutions to the system is a: in :
plane in R^5
A homogeneous linear system is always consistent
true
Any set of n linearly independent vectors in R^n is a basis for R^n.
true
Consider the subspace W of R 4 given by: W= {(x y z w) in R4 | x-y-z+w=0} Then dim(W) =3
true
For any matrix A , we have the equality 2A + 3A =5A
true
For any matrix A, there exists a matrix B so that A + B=0
true
If A is a 3 × 8 matrix, then dim(Nul A) > dim(Col A)
true
If A is a 30 × 20 matrix and dim(Col A) = 10, then the null space of A is a 10- dimensional subspace of R^20
true
If A is an m x n matrix and B is an n x m matrix then AB and BA are both defined
true
If A is an m×n matrix and m > n, then then there is at least one vector b in R^m which is not in the span of the columns of A.
true
If A is an n × n matrix and the equation Ax = b has at least one solution for each b in R^n , then A must be invertible
true
If T : R n → R m is a linear transformation, then the zero vector must be a solution to the equation T(x) = 0
true
If the bottom row of a matrix in reduced row echelon form contains all 0s to the left of the vertical bar and a nonzero entry to the right, then the system has no solution.
true
If v 1 and v 2 are vectors in R^n, then the vector 3v 1-v2 is in Span {v1,v2}
true
If v1, v2, and v3 are vectors in R^2, then the vector equation x1v1+x2v2+x3v3=(0 0) (as in a vector) must have infinitely many solutions
true
If {v1 , . . . , vp } is a linearly independent set of vectors in R n , then p ≤ n.
true
If {v1,v2,v3,v4} is a basis for R^4, then {v1,v2,v3} must be linearly independent
true
Let A be the 2 × 2 matrix that rotates vectors in R 2 by 65 degrees counterclockwise. Then A has no real eigenvalues
true
Suppose A is a 2 × 2 matrix and b is a vector in R^2 . If the solution set to Ax = b is the span of (-3 1), then b = (0 0)
true
Suppose A is a 3 x 2 matrix and b is a vector in R^3 so that Ax=b has exactly one solution. Then the only solution to the homogeneous equation Ax=0 is the trivial solution
true
Suppose A is a 3 × 2 matrix whose columns are linearly independent, and let T be the matrix transformation T(x) = Ax. Then {T(1 0), T(0 1)} is a basis for the range of T
true
Suppose A is an 4×3 matrix whose first column is the sum of its second and third columns. Then the equation Ax = 0 has infinitely many solutions.
true
Suppose A is an n × n matrix and Ax = 0 has only the trivial solution. Then each b in R n can be written as a linear combination of the columns of A
true
Suppose a system of linear equations corresponds to an augmented matrix whose RREF has bottom row equal to ( 0 1 0 | 0)
true
T is a one-to-one transformation from R3 to R3 means for every y in R3, there is at most one x in R3 such that T(x)=y
true
T is a transformation from R3 to R3 means for every x in R3, there is a y in R3 such that T(x)=y
true
T is an onto transformation from R3 to R3 means for every y in R3, there is an x in R3 such that T(x)=y
true
The column space of an m x n matrix is a subspace of R^m
true
The columns of a matrix with dimensions m×n, where m<n, must be linearly dependent
true
The equation 3x+ln(2)y=pi is a linear equation in x and y
true
The equation Ax=b is homogenous if the zero vector is a solution
true
The following vector equation is consistent for every b in R^3 : x1(1 0 3) +x2(0 1 0) + x3(0 2 3)=b
true
The set of all solutions of m homogeneous equations in n unknowns is a subspace of R^n
true
The solution set of a consistent inhomogeneous system Ax=b is obtained by translating the solution set of Ax=0
true
Two vectors are linearly dependent if and only if they are collinear
true
Complete the following mathematical definition of linear combination: Let v1, v2, ...vp, and w be vectors in R^n. We say w is a linear combination of v1, ..., vp if...
w=x1v1+x2v2+...+xpvp for some scalars x1,...xp
Is T: R^2 ---> R^2 rotating counterclockwise by 15 degrees one-to-one?
yes
Suppose that the plane x1−4x2+ x3 = 0 is the set of solutions to the matrix equation Ax = 0, and suppose that (-3 1 2) is a solution to Ax= (2 1 0) i) is it true that A(4 1 0) = (0 0 0)? ii) is it true that (1 2 2) is a solution to Ax= (2 1 0) ?
yes yes
Ax=0 has only trivial solution vs. Ax=0 has a nontrivial solution
zero, pivot in every column vs. nonzero, there is a free variable or a column with no pivot