Linear Algebra Master True and False
If the columns of an m×n matrix A span Rm then the equation Ax=b is consistent for each b in Rm.
True
If the equation Ax=0 has a nontrivial solution, then A has fewer than n pivot points.
True
If y is in a subspace WW as well as its orthogonal complement W⊥, then y must be the zero vector.
True
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
There are exactly three vectors in the set {a1,a2,a3}
True
Two vectors are linearly dependent if and only if they are colinear.
True
Suppose that AA is a 7 ×5 matrix which has a null space of dimension 4. What is the rank?
1(dimension of the column space)
If the linear transformation T(x)=Ax is one-to-one, then the columns of A form a linearly dependent set.
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
If {v1,v2,v3} is an orthonormal basis for W, then multiplying v3 by a scalar cc gives a new orthonormal basis {v1,v2,cv3}
False
Matrices with the same eigenvalues are similar matrices.
False
Row operations on a matrix do not change its eigenvalues.
False
Span{a1,a2}contains only the line through a1a1 and the origin, and the line through the a2a2 and the origin.
False
Suppose A and B are invertible matrices: (A+B)2=A2+B2+2AB
False
Suppose A and B are invertible matrices: (AB)−1=A−1B−1
False
Suppose A and B are invertible matrices: A+B is invertible.
False
The homogeneous system Ax=0 has the trivial solution if and only if the system has at least one free variable.
False
Col(A)⊥=Nul(AT)=Nul[A]
Importante
Nul(A)⊥=Row(A)=Col(AT)=Col[a]
Importante
If the equationAx=0 has the trivial solution, then the columns of A span Rn.
Maybe,False
The transformation T defined by T(x1,x2)=(4x1−2x2,3|x2|) Is this a Linear transformation?
No
The transformation T defined by T(x1,x2,x3)=(x1,0,x3)T(x1,x2,x3)=(x1,0,x3) Is this a Linear transformation?
Yes
A is diagonalizable if and only if A has n eigenvalues, counting multiplicity.
False
A least-squares solution of Ax=b is a vector x̂ such that ∥b−Ax∥≤∥b−Ax̂ ∥ for all xx in RnRn.
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
A determinant of an n×n matrix can be defined as a sum of multiples of determinants of (n−1)×(n−1) submatrices.
True
A homogeneous system is always consistent.
True
A is invertible if and only if 0 is not an eigenvalue of A.
True
A matrix that is similar to the identity matrix is equal to the identity matrix.
True
A number cc is an eigenvalue of A if and only if (A−cI)v=0 has a nontrivial solution.
True
Asking whether the linear system corresponding to an augmented matrix [a1a2a3b] has a solution amounts to asking whether bb is in Span{a1,a2,a3}
True
For any matrix A, there exists a matrix B so that A+B=0
True
For any matrix A, we have the equality 2A+3A=5A
True
If A is diagonalizable, then A2 is also diagonalizable
True
If A is invertible, then the equation Ax=b has exactly one solution for all b in Rn.
True
If A is n×n and A has n distinct eigenvalues, then the corresponding eigenvectors of A are linearly independent.
True
If A is the matrix of an orthogonal projection, then A2=A.
True
If A=QR ,where Q has orthonormal columns, then R=QTA.
True
If AT is row equivalent to the n×n identity matrix, then the columns of AA span Rn.
True
If W=Span{x1,x2,x3} with {x1,x2,x3} linearly independent, and if {v1,v2,v3} is an orthogonal set of nonzero vectors in W, then {v1,v2,v3} is a basis for W.
True
If a matrix is in reduced row echelon form, then the first nonzero entry in each row is a 1 and has 0s below it.
True
If bb is in the column space of A, then every solution of Ax=b is a least-squares solution.
True
If the characteristic polynomial of a 2×2 matrix is λ2−5λ+6, then the determinant is 6.
True
If the linear transformation T(x)=Axis onto, then it is also one-to-one.
True
If the transpose of A is not invertible, then A is also not invertible.
True
If there is a basis of Rn consisting of eigenvectors of A, then A is diagonalizable.
True
If vv is an eigenvector of A, then cv is also an eigenvector of A for any number c≠0
True
If xx is not in a subspace W, then x−projW(x) is not zero.
True
If x̂ is the least-squares solution of Ax=b, then Ax̂ is the point in the column space of A closest to b
True
If x̂ is the least-squares solution of Ax=b, then Ax̂ is the point in the column space of A closest to b.
True
If y is in a subspace W, then the orthogonal projection of y onto W is y.
True
Suppose A and B are invertible matrices: (In−A)(In+A)=In−A2
True
Suppose A and B are invertible matrices: A7 is invertible.
True
The (i,j) minor of a matrix A is the matrix Aij obtained by deleting row ii and column j from A.
True
The Gram-Schmidt process produces from a linearly independent set {x1,...,xp} an orthogonal set {v1,...,vp} with the property that for each k, the vectors v1,...,vk span the same subspace as that spanned by x1,...,xk
True
The absolute value of the determinant of A equals the volume of the parallelepiped determined by the columns of A.
True
The cofactor expansion of det A along the first row of AA is equal to the cofactor expansion of det A along any other row.
True
The column space of an m×n matrix is a subspace of Rm.
True
The columns of a matrix with dimensions m×n, where m<n, must be linearly dependent.
True
The columns of an invertible n×n matrix form a basis for Rn
True
The product of any two invertible matrices is invertible.
True
The solution set of a linear system whose augmented matrix is [a1a2a3∣b] is the same as the solution set of Ax=b, if A=[a1a2a3]
True
The transformation TT defined by T(x1,x2,x3)=(x1,x2,−x3) Is this a Linear transformation?
Yes
Null space &Column Space are always subspaces..but
Check dimensions if a subspace in R3 idk
If its says Column Space use span{xx} if it says basis don't say span
Don't be a dumbas*
A square matrix with two identical columns can be invertible.
False
For any matrices A and B, if the product AB is defined, then BA is also defined.
False
If A is a 5×4 matrix, and B is a 4x3 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 diagonalizable, then A is invertible.
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
If a linear system has four equations and seven variables, then it must have infinitely many solutions.
False
If an n×n matrix A has fewer than n distinct eigenvalues, then A is not diagonalizable.
False
If the augmented matrix [A∣b] has a pivot position in every row, then the equation Ax=b is inconsistent.
False
The null space of an m×n matrix is a subspace of Rm.
False
There are exactly three vectors in Span{a1,a2,a3}
False
λ is an eigenvalue of a matrix A if A−λI has linearly independent columns.
False
The eigenvalues of the matrix of an orthogonal projection are −1 and 1.
False, 0 and 1
If SS is a set of linearly dependent vectors, then every vector in Scan be written as a linear combination of the other vectors in S.
False, Atleast one of the vectors are linearly independent not all
The columns of matrix A are linearly independent if the equation Ax=0 has the trivial solution.
False, has ONLY
The eigenvalues of AA are the entries on its main diagonal.
False, triangle or diagonal
The determinant of a triangular matrix is the sum of the entries of the main diagonal.
False,Product
. The equation Ax=b is consistent if the augmented matrix [A∣b] has a pivot position in every row.
False,column
The transformation T defined by T(x1,x2,x3)=(1,x2,x3) Is this a Linear transformation?
No
If A is an m×n matrix and if the equation Ax=b is inconsistent for some b in Rm, then A cannot have a pivot position in every row.
True
If A is an m×n matrix then ATA and AAT are both defined.
True
If A is an m×n matrix whose columns do not span Rm, then the equation Ax=b is inconsistent for some b in Rm.
True
If b is in the column space of A, then every solution of Ax=b is a least-squares solution.
True
The equation Ax=b is homogenous if the zero vector is a solution. -Review Homogeoulity
True
The general least-squares problem is to find an x that makes Ax as close as possible to b.
True
The set of all solutions of a system of mm homogeneous equations in n unknowns is a subspace of Rn.
True
The solution set of a consistent inhomogeneous system Ax=b is obtained by translating the solution set of Ax=0. ...why
True
Column Space
Rm
Null Space
Rn
For every x in R3, there is a y in R3 such that T(x)=y
T is a function from R3 to R3
For every y in R3, there is at most one x in R3 such that T(x)=y
T is a one-to-one function from R3 to R3
For every y in R3, there is a x in R3 such that T(x)=y
T is an onto function from R3 to R3
