Linear Algebra Master True and False

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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


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