Linear Algebra Final
Suppose that a system of three equations with eleven unknowns corresponds to a matrix in row echelon form. What is the largest possible number of pivots for this matrix?
3 Pivots
Determine whether or not T is one-to-one in each of the following situations: n < m
Could be either one to one or onto.
Determine whether or not T is one-to-one in each of the following situations: n=m
Could be either one to one or onto.
Assume A is an n×n matrix. If det A is zero, then two columns of A must be the same, or all of the elements in a row or column of A are zero.
False
Assume A is an n×n matrix. If the columns of A are linearly independent, then det A=0.
False
Assume A is an n×n matrix. The determinant of a triangular matrix is the sum of the entries of the main diagonal.
False
Assume A is an n×n matrix. det(A+B)=det(A)+det(B)
False
Suppose A and B are invertible matrices. (A+B)^2=A^2+B^2+2AB. True or False?
False
Suppose A and B are invertible matrices. (AB)^−1=A^−1B^−1. True or False?
False
Suppose A and B are invertible matrices. A+B is invertible. True or False?
False
Suppose A is a square matrix. The eigenvalues of A are the entries on its main diagonal.
False
Suppose A is an n×n matrix. A is diagonalizable if and only if A has n real eigenvalues, counting multiplicity.
False
Suppose A is an n×n matrix. If A is diagonalizable, then A is invertible.
False
Suppose A is an n×n matrix. If an n×n matrix A has fewer than n distinct real eigenvalues, then A is not diagonalizable.
False
Suppose A is an n×n matrix. True or False? A square matrix with two identical columns can be invertible.
False
Suppose A is an n×n matrix. True or False? If the linear transformation T(x)=Ax is one-to-one, then the columns of A form a linearly dependent set.
False
Suppose a1, a2, and a3 are three different nonzero vectors. True or False: Span{a1,a2} contains only the line through a1 and the origin, and the line through the a2 and the origin.
False
Suppose a1, a2, and a3 are three different nonzero vectors. True or False: There are exactly three vectors in Span{a1,a2,a3}.
False
True or False: If a linear system has four equations and seven variables, then it must have infinitely many solutions.
False
True or 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
True or False: If a system of linear equations has more variables than equations, then it must have infinitely many solutions.
False
True or 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
True or False: If an augmented matrix is in reduced row echelon form, then it must have a pivot in its left most column.
False
True or False: If x is a nontrivial solution of Ax=0, then every entry of x is nonzero.
False
True or False: Suppose A is an n x n matrix and λ is an eigenvalue of A. Then the column space of A-λIn is the λ-eigenspace for the matrix A.
False
True or False: The homogeneous system Ax=0 has the trivial solution if and only if the system has at least one free variable.
False
True or False: The plane 2x+y−z=3 contains the point (1,1,1).
False
True or False: The set of all solutions (x,y,z) to the equation x−y−z=0 is a line in R3.
False
True or False? Suppose A is a 3×3 matrix and λ is a real number with the property that the equation Ax=λx is satisfied by some nonzero vector x. A is not invertible.
False
True or False? Suppose A is a 3×3 matrix and λ is a real number with the property that the equation Ax=λx is satisfied by some nonzero vector x. A−λ is invertible.
False
True or false? Row operations on a matrix do not change its eigenvalues.
False
True or false? The characteristic polynomial of the zero matrix is 0.
False
True or false? λ is an eigenvalue of a matrix A if A−λI has linearly independent columns.
False
True or False: The equation Ax=b has a solution for all b precisely when it is a square matrix.
False, the equation Ax=0 always admits the trivial solution, whether or not the columns of A are linearly independent.
True or False: If B is an echelon form of a matrix A, then the pivot columns of B form a basis for the column space of A.
False. A basis of the column space of A consists of the columns of A that correspond to the pivot columns in B.
True or False: The null space of an m×n matrix is a subspace of R^m.
False. The null space of an m×n matrix A is the space of all solutions to the matrix equation Ax=0. A solution of this equation must be a vector in Rn, so the null space is a subspace of Rn.
True or 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: in order for S to be linearly dependent, only one vector in S need be expressible as a linear combination of the others.
What is the 2x2 matrix for the reflection about y-axis?
First row: ( -1 0 ) Second row: ( 0 1 )
What is the 2x2 matrix for Counter-clockwise rotation by π/2 radians?
First row: ( 0 -1 ) Second row: ( 1 0 )
What is the 2x2 matrix for the clockwise rotation by π/2 radians?
First row: ( 0 1 ) Second Row: ( -1 0 )
What is the 2x2 matrix for the reflection about the x-axis?
First row: ( 1 0 ) Second row: ( 0 -1 )
What is the 2x2 matrix for the projection onto the x-axis given by T(x,y) = (x, 0)?
First row: ( 1 0 ) Second row: ( 0 0 )
What is the 2 x2 matrix for the reflection about the line y=x?
First row: (0 1) Second row: (1 0)
Suppose A is an n×n matrix. True or False? If the equation Ax=0 has the trivial solution, then the columns of A span Rn.
Maybe/False
Determine whether or not T is one-to-one in each of the following situations: n>m
T is not a one-to-one transformation.
Let A be an m×n matrix with associated transformation T(x)=Ax, if The row-echelon form of A has a row of zeros is it onto, not onto, or need more information?
T is not onto
Let A be an m×n matrix with associated transformation T(x)=Ax, if Two rows in the row-echelon form of A do not have pivots then is it onto, not onto, or need more information?
T is not onto
Let A be an m×n matrix with associated transformation T(x)=Ax, if Every row in the row-echelon form of A has a pivot is it onto, not onto, or need more information?
T is onto
Suppose that A is a 3×7 matrix that has an echelon form with one zero row. Find the dimension of the column space of A, and the dimension of the null space of A. What is the dimension of the column space of A?
The dimension of the column space A is 2.
Suppose that A is a 3×7 matrix that has an echelon form with one zero row. Find the dimension of the column space of A, and the dimension of the null space of A. What is the dimension of the Null space of A?
The dimension of the null space is 5.
Assume A is an n×n matrix. A determinant of an n×n matrix can be defined as a sum of multiples of determinants of (n−1)×(n−1) submatrices.
True
Assume A is an n×n matrix. A row replacement operation does not affect the determinant of a matrix.
True
Assume A is an n×n matrix. If two columns of A are the same, then the determinant of that matrix is zero.
True
Assume A is an n×n matrix. The (i,j) minor of a matrix A is the matrix Aij obtained by deleting row i and column j from A.
True
Assume A is an n×n matrix. The absolute value of the determinant of A equals the volume of the parallelepiped determined by the columns of A.
True
Assume A is an n×n matrix. The cofactor expansion of det A along the first row of A is equal to the cofactor expansion of det A along any other row.
True
If T is a one to one transformation from R^3 -> R^2, then for every y in R^3, there is at most one x in R^3 such that T(x)=y.
True
If T is a transformation from R3 to R3, then for every x in R3, there is a y in R3 such that T(x)=y.
True
If T is an onto transformation from R^3 to R^3, then for every y in R3, there is at least one x in R3 such that T(x)=y.
True
Let W be the subspace of R^3 that consists of all vectors satisfying 2x-y+z =0 True or False: if u and v are linearly independent vectors in W, then {u,v} must be a basis for W.
True
Suppose A and B are invertible matrices, (In−A)(In+A)=In−A^2. True or False?
True
Suppose A and B are invertible matrices. A^7 is invertible. True or False?
True
Suppose A is a square matrix. A is invertible if and only if 0 is not an eigenvalue of A.
True
Suppose A is a square matrix. A number c is an eigenvalue of A if and only if (A−cI)v=0 has a nontrivial solution.
True
Suppose A is a square matrix. If A is n×n and A has n distinct eigenvalues, then the corresponding eigenvectors of A are linearly independent.
True
Suppose A is a square matrix. If v is an eigenvector of A, then cv is also an eigenvector of A for any number c≠0.
True
Suppose A is an n×n matrix. If A is diagonalizable, then A^2 is also diagonalizable.
True
Suppose A is an n×n matrix. If there is a basis of R^n consisting of eigenvectors of A, then A is diagonalizable.
True
Suppose A is an n×n matrix. If −A is not invertible, then A is also not invertible. True or False?
True
Suppose A is an n×n matrix. True or False? If A is invertible, then the equation Ax=b has exactly one solution for all b in R^n.
True
Suppose A is an n×n matrix. True or False? If A^2 is row equivalent to the n×n identity matrix, then the columns of A span R^n.
True
Suppose A is an n×n matrix. True or False? If the equation Ax=0 has a nontrivial solution, then A has fewer than n pivots.
True
Suppose A is an n×n matrix. True or False? If the linear transformation T(x)=Ax is onto, then it is also one-to-one.
True
Suppose A is an n×n matrix. True or False? The product of any two invertible matrices is invertible.
True
Suppose a1, a2, and a3 are three different nonzero vectors. True or False: Asking whether the linear system corresponding to an augmented matrix [a1a2a3b] has a solution amounts to asking whether b is in Span{a1,a2,a3}.
True
Suppose a1, a2, and a3 are three different nonzero vectors. True or False: The solution set of the linear system whose augmented matrix [a1a2a3b] is the same as the solution set of the equation x1a1+x2a2+a3x3=b.
True
Suppose a1, a2, and a3 are three different nonzero vectors. True or False: There are exactly three vectors in the set {a1,a2,a3}.
True
True or False: A homogeneous linear system is always consistent.
True
True or False: If A and B are invertible matrices, then det(AB) does not equal 0.
True
True or False: If A is an n x n matrix and the columns of A are linearly independent, then the matrix transformation T(x) = Ax is one to one and onto.
True
True or False: If K is any real real number, then span{(1/k), (0 / -1)} must be R^2
True
True or False: 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
True or False: If the reduced row echelon form of an augmented matrix has bottom row equal to (0 1 2 | 3), then the corresponding system of linear equations must have infinitely many solutions.
True
True or False: If the solution to a system of linear equations is given by (4−2z,−3+z,z), then (4,−3,0) is a solution to the system.
True
True or False: Let A be a matrix with linearly independent columns, then the equation Ax=b has a solution for all b precisely when it is a square matrix.
True
True or False: Let A be a matrix with more rows than columns, then the columns of A could be either linearly dependent or linearly independent.
True
True or False: Suppose A is a 3 x 3 matrix with characteristic polynomial det(A-λI) = -λ^3-λ^2+λ+1, Then A must be invertible.
True
True or False: The equation 3x+ln(2)y=π is a linear equation in x and y.
True
True or False: The equation Ax=b is homogenous if the zero vector is a solution.
True
True or False; The solution set of a consistent inhomogeneous system Ax=b is obtained by translating the solution set of Ax=0.
True
True or False? Suppose A is a 3×3 matrix and λ is a real number with the property that the equation Ax=λx is satisfied by some nonzero vector x. A−λI is not invertible.
True
True or false? If A is a 4×4 matrix with characteristic polynomial λ4+λ3+λ2+λ, then A is not invertible.
True
True or false? If the characteristic polynomial of a 2×2 matrix is λ2−5λ+6, then the determinant is 6.
True
True or False: The column space of an m×n matrix is a subspace of R^m.
True. By definition, the column space is the span of the columns, and any span is a subspace.
True or False: Any set of n linearly independent vectors in R^n is a basis for R^n.
True. Since Rn has dimension n, we know from the Basis Theorem that any set of n linearly independent vectors Rn will form a basis of Rn.
True or False: The set of all solutions of a system of m homogeneous equations in n unknowns is a subspace of R^n.
True. To solve a system of m equations in n unknowns, we can insert the equations as rows of a m×n matrix, denoted A, and solve the matrix equation Ax=0. The solution set is the null space of A, which is a subspace of Rn.
True or False: Two vectors are linearly dependent if and only if they are collinear.
True: if ax+by=0, with a≠0 (for instance), then x=−bay, which says that x and y lie on the same line. Conversely, if x and y lie on the same line, then there exists a≠0 such that x=ay (unless y=0, in which case swap x and y); then ay−x=0 is an equation of linear dependence.
True or False: The columns of a matrix with dimensions m×n, where m<n, must be linearly dependent.
True: it is impossible for such a matrix to have a pivot in each column. Alternatively, such a matrix must give rise to at least one free variable.
Suppose A is a 3 x 4 matrix and T is the corresponding matrix transformations T(x) = Ax. Can T be one to one? Can T be onto?
Yes, one to one! No, not onto!
Let T be a one-to-one matrix transformation from Rn to Rm. What can one say about the relationship between m and n?
n is less than or equal to m.
