MATH 221

(1 point) If A and B are 3×3 matrices, det(A)=−3, det(B)=4, then det(AB)= , det(3A)= det(A^T)= det(B−1)= det(B3)= .

-12 , -81, -3, 1/4, 64

Suppose A is a real 4×4 matrix with eigenvalues 3, 4, and 9i. What is the fourth eigenvalue?

-9i . Complex eigenvalues come in conjugate pairs, and the eigenvectors corresponding to conjugate complex eigenvalues are also conjugates.

(1 point) Supppose A is an invertible n×n matrix and v⃗ is an eigenvector of A with associated eigenvalue −6. Convince yourself that v⃗ is an eigenvector of the following matrices, and find the associated eigenvalues. The matrix A^4 has an eigenvalue The matrix A^−1 has an eigenvalue The matrix A+5In has an eigenvalue The matrix −7A has an eigenvalue

1) 1296 2)1/-6 3) -1 4) 42

True or False Suppose A is a square matrix. Select "True" if the statement is always true, and "False" otherwise. 1)The eigenvalues of A are the entries on its main diagonal. 2)A number c is an eigenvalue of A if and only if (A−cI)v=0 has a nontrivial solution. 3)A is invertible if and only if 0 is not an eigenvalue of A. 4)If v is an eigenvector of A, then cv is also an eigenvector of A for any number c≠0. 5)If A is n×n and A has n distinct eigenvalues, then the corresponding eigenvectors of A are linearly independent.

1) False . This is only the case if AA is a triangular or diagonal matrix. 2) True. True. By definition, c is an eigenvalue if and only if there exists some nonzero vector v such that Av=cv⟺Av−cv=0⟺(A−cI)v=0. 3) True. Zero is an eigenvalue if and only if Ax=0xAx=0x has a nontrivial solution, if and only if Ax=0Ax=0 has a nontrivial solution. By the invertible matrix theorem, this is equivalent to the non-invertibility of A. 4)True. If Av=λv then A(cv)=c(Av)=c(λv)=λ(cv), with λ being the eigenvalue corresponding to v. Alternatively, the λ-eigenspace is a subspace, so any nonzero multiple of a nonzero vector in the λ-eigenspace is again an eigenvector with eigenvalue λ. 5). True eigen vectors with distinct eigen values are linearly independent.

Properties of Onto

1) Range of T is equal to the codomain of T i.e. m 2) every vector in the codomain is the output of some input vector

Let T be an linear transformation from Rr to Rs. Let A be the matrix associated to T. Fill in the correct answer for each of the following situations. 1)Two columns in the row-echelon form of A are not pivot columns. 2)The row-echelon form of A has no column corresponding to a free variable 3)Every column in the row-echelon form of A is a pivot column. 4)The row-echelon form of AA has a column corresponding to a free variable.

1) T is not one-to-one 2) T is one-to-one 3) T is one-to-one 4) T is not one-to-one

True or False 1)If A is diagonalizable, then A2 is also diagonalizable. 2)If an n×n matrix A has fewer than n distinct eigenvalues, then A is not diagonalizable. 3)A is diagonalizable if and only if A has n eigenvalues, counting multiplicity. 4)If there is a basis of Rn consisting of eigenvectors of A, then A is diagonalizable. 5)If A is diagonalizable, then A is invertible.

1) True. If A is diagonalizable, then there exists a diagonal matrix D and an invertible matrix C such that A=CDC−1. Then A2=(CDC−1)2=CD2C−1, where D2 is a diagonal matrix. Therefore, A2 is diagonalizable. 2) False. For instance, [10;01] is diagonal, but its only eigenvalue is 1. 3) False. For instance, [10;11] has one eigenvalue 1 of multiplicity 2, but it is not diagonalizable, because the 1-eigenspace only has dimension 1. 4) True True. If C is an n×n matrix whose columns are linearly independent eigenvectors, then A=CDC^−1, where D is the diagonal matrix whose diagonal entries are the corresponding eigenvalues, in the same order. 5) False. For instance, [1000] is diagonal but not invertible.

Echelon Form

1) all zero rows at bottom 2) first non zero entry of every row is a 1 3) below first non zero entry of a row all entries are zero

True or False 1)If the characteristic polynomial of a 2×2 matrix is λ2−5λ+6, then the determinant is 6. 2)Matrices with the same eigenvalues are similar matrices. 3)Row operations on a matrix do not change its eigenvalues. 4)λ is an eigenvalue of a matrix A if A−λI has linearly independent columns. 5)A matrix that is similar to the identity matrix is equal to the identity matrix.

1)True. Let A=[ac bd]. Then det(A−λI)=(a−λ)(d−λ)−bc=λ2−(a+d)λ+(ad−bc); the constant coefficient is ad−bc=det(A). 2) False. Similar matrices do have the same eigenvalues; however, [1011] and [1001] are not similar (the only matrix similar to the identity matrix is the identity matrix itself), and they both only have 1 as an eigenvalue. 3) False. For instance, the invertible matrix [2002] is row equivalent to the identity matrix, but it does not have 1 as an eigenvalue. 4) False. For λ to be an eigenvalue of the matrix A, the matrix equation (A−λI)x=0 must have more than just the trivial solution, by definition. Thus A−λI must have linearly dependent columns. 5) True. By definition, if a matrix A is similar to the identity matrix then there exists an invertible matrix P such that A=PIP−1. But PI=P for any P, so A=PP−1=I.

properties of one-to-one transformation

1. T is one to one 2. For every b in Rm the equation T(x)=b has at most one solution 3. For every b in Rm, the equation Ax=b has a unique solution or is inconsistent 4 A(x)=0 has only the trivial solution 5. The columns of A are linearly independent 6. A has a pivot in every column 7. The range of T has dimension n

1 point) Are the following statements true or false? 1. If u and v are in a subspace S, then every point on the line connecting u and v is also in S. [The line is the set of vectors you can form as tu+(1−t)v for different values of t] 2. If T:R4→R10 is a linear transformation, then range (T) (also known as the image of T) is a subspace of R10. 3. The intersection of two subspaces of Rn forms another subspace of Rn. 4. The sum of two subspaces of Rn forms another subspace of Rn. The sum of V and W means the set of all vectors v⃗ +w⃗ where v⃗ is an element of V and w⃗ is an element of W.

1. True 2. True 3. True 4. True

Assume A is an n×n matrix. Answer "True" if the statement is always true, and "False" otherwise. 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. det(A+B)=det(A)+det(B) A row replacement operation does not affect the determinant of a matrix. If the columns of A are linearly independent, then det A=0. If two columns of A are the same, then the determinant of that matrix is zero.

1. false 2. false 3. true 4. false 5. true explanations False. This is not necessarily the case: consider the matrix [1224], which is not invertible because its determinant is 1⋅4−2⋅2=0. Its columns are not the same (although they are linearly dependent), and none of the entries in any column or row is 0. False. This is usually not true: consider the matrices A=[1−2−11] and B=[−121−1]. We note that A+B is the 2×2 zero matrix, which has determinant 0. On the other hand det(A)=det(B)=−1, so det(A)+det(B)=−2. True. False. If the columns of a matrix are linearly independent, then by the invertible matrix theorem, this means the matrix is invertible, which in turn implies that its determinant is non-zero. True. If a matrix has two columns which are the same, then its columns are linearly dependent. By the invertible matrix theorem, the matrix is not invertible, and thus its determinant must be 0.

parametric form

1. write the system as an augmented matrix 2. row reduce to rref 3. write the corresponding solved system of linear equations 4. move all the free variables to the right hand side of the equations

In a 5x5 matrix If A has two pivots, then the dimension of NulA is 22.

A has two pivots means the dimension of the column space of A. By the Rank Theorem, the dimension of the null space is 5−2=3

consistency and span

A vector is in the span of v1, v2,.....,vk is the same as saying that the vector equation x1v1+x2v2.....+xkvk = b is consistent i.e. it has at least one solution

properties of vector product

A(u+v) = Au + Av A(cu) = cAu where u v are vectors and c is a scalar

homogenous system of linear equations

Ax=0 when b not 0 inhomogenous

algebraic consistency

Ax=b

Null space calculation

Compute the parametric vector form of the solutions to the homogeneous equation Ax=0. The vectors attached to the free variables form a spanning set for Null(A).

Which of these transformations are invertible? A. Trivial transformation (i.e. T(v⃗ )=0⃗ for all v⃗ ) B. Dilation by a factor of 5 C. Identity transformation (i.e. T(v⃗ )=v⃗ for all v⃗ ) D. Reflection in the x-axis E. Projection onto the z-axis F. Rotation about the y-axis

Dilation by a factor of 5 Identity Transformation Reflection in the x-axis Rotation about the y-axis

A 5×5 real matrix has an even number of real eigenvalues.

False. A 5×5 real matrix can only have an odd number of real eigenvalues. This is because if it has an eigenvalue of the form a+bi, then it necessarily has another eigenvalue a−bi, so the number of complex eigenvalues is always even. Since a 5×5 real matrix has 5 eigenvalues (counted with multiplicity), and it has an even number of complex eigenvalues, it must have an odd number of real eigenvalues.

There exists a real 2×2 matrix with the eigenvalues i and 2i.

False. Complex eigenvalues come in conjugate pairs, so if a 2×2 matrix has i as an eigenvalue, its other eigenvalue must be −i.

Rank Theorem

If A is a matrix with n columns then rank(A) + nullity(A) = n

Dimension

Let V be a subspace in Rn. the number of vectors in any basis of V is called the dimension of V and is written as dim V.

spans and consistency

Matrix equation Ax=b has a solution if b is in the span of the columns of A

In a 5 by 5 If A has three pivots, then ColA is a (two-dimensional) plane

No. The dimension of the column space is the same as the number of pivot columns, so it must be 3

not onto

Range of T is smaller than codaim of T There exists a vector b in Rm such that T(x)=b has no solution There is a vector in the codomain that is not the output of any input vector

Rank

Rank of the matrix is dimension of the column space Col(A). number of columns with pivots.

Null Space

Subspace of Rn consisting of all the solutions to the homogeneous equation Ax=0 Null(A) ={x in Rn | Ax=0}

if a is an m×n non augmented matrix T or F then the following are equivalent 1) Ax=b has a solution for all b in Rm 2) The span of the columns of A is all of Rm 3) A has a pivot position in every row

T T T

One to one transformation

T(x1) /= T(x2) for every vector b in Rm the equation T(x)=b has at most one solution x in Rn

Onto Transformation

T:Rn->Rm is onto if for every vector b in Rm, the equation T(x)=b has at least one solution x in Rn

Find a 3×3 matrix A such that Ax=5x for all x in R3

The 3×3 identity matrix I3 I3x=xI3x=x for all xx in R3R3. Therefore, if we take A=5I3=⎡⎣⎢500050005⎤⎦⎥A=5I3=[500050005], then Ax=5I3x=5xAx=5I3x=5x for all xx in R3R3.

Let T:R2→R2 be the linear transformation that first rotates points clockwise through π/6 and then reflects points through the line y=x. Find the standard matrix A for T. (Your answer can be in terms of trigonometric functions and pi.)

The columns of AA are T(e1)T(e1) and T(e2)T(e2). Rotating e1 clockwise through π/6 gives (cos(−π/6),sin(−π/6)) flipping over y=x gives T(e1)=(sin(−π/6),cos(−π/6))T(e1)=(sin⁡(−π/6),cos⁡(−π/6)). Rotating e2 clockwise through π/6 gives (−sin(−π/6),cos(−π/6)); flipping over y=x gives T(e2)=(cos(−π/6),−sin(−π/6))T(e1)=(cos⁡(−π/6),−sin⁡(−π/6)). It follows that A=[−sin(π/6)cos(π/6) cos(π/6)sin(π/6)]

Dimension of the solution set

The number of free variables is the dimension of the solution set. E.g. when there is one free variable the solution set is a line when there are two free variables the solution set is a plane etc.

Dimension of Col(A)

The number of pivot columns in rref

Suppose a matrix A has columns that span Rn. choose the best statement: A. Then the equation Ax=0 must have nontrivial solutions. B. Then the equation Ax=0 cannot have nontrivial solutions. C. Then the equation Ax=0 will have nontrivial solutions precisely when it is not square. D. Then the equation Ax=0 will have nontrivial solutions only if one column is a multiple of another column. E. Then the equation Ax=0 can have nontrivial solutions, but the shape of the matrix will not give us that information. F. none of the above

Then the equation Ax=0 will have nontrivial solutions precisely when it is not square.

A nonzero subset has infinitely many different bases, but they all contain the same number of vectors

True

A subspace is a span

True

Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x.

True

Any two non collinear vectors form a basis of R^2

True

If A is an m × n matrix and if the equation Ax=b is inconsistent for some bb in Rm, then A cannot have a pivot position in every row.

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 the equation Ax=b is inconsistent, then b is not in the set spanned by the columns of A.

True

If u,v are vectors in V and c and d are scalar, then cu and dv are also in V and cu+dv are also in V

True

If v is a vector in subspace V then all scalar multiples of v are also in V

True

if v1,v2,.......,vn are all in V then the Span {v1,v2.....vn} is also in V

True

wide matrices do not have one-to-one transformations

True. A associated with T has m rows and n columns in order for A to have a pivot in every column it must have atleast as many rows as columns so n<= m

Real eigenvalues of a real matrix always correspond to real eigenvectors. T or F.

True. If λλ is a real eigenvalue of AA, then the equation (A−λI)x=0(A−λI)x=0 has a nonzero real solution.

Every real 3×3 matrix must have a real eigenvalue.

True. The characteristic polynomial has degree 3, and any degree 3 real polynomial has at least one real root.

Basis for the null space

Vectors attached to the free variables in parametric form

When does Ax=0 have a non-trivial solution?

When A has a column without a pivot position

when there are no free variables what is the null space?

When there are no free variables the only solution to Ax=0 is the trivial solution. I.e. Nul(A)={0}=span{0}

If Ax=0 has only the trivial solution, then ColA=R5

Yes. Ax=0 has only the trivial solution means that all the columns of A are linearly independent vectors in R5, and since there are 5 of them, they form a basis for R5

Suppose B is a real 2×2 matrix with two complex eigenvalues. If [2+i;3−2i] is an eigenvector corresponding to one of the eigenvalues, then what is an eigenvector corresponding to the other?

[2-i;3+2i].Complex eigenvalues come in conjugate pairs, and the eigenvectors corresponding to conjugate complex eigenvalues are also conjugates.

product of a row vector of length m and a column vector of length n

[a1 a2 a3][x1; x2; x3] a1 x1 + a2 x2 etc

linear dependence relation

a homogeneous vector equation where the weights are all specified and at least one weight is nonzero

row vector

a matrix with one row

linearly independent

a set of vectors is linearly independent is there exists x1 x2 to xk mot all equal to zero such that their linear combination is equal to zero.

unique solution

all pivot columns except the augmented column

rref of an inconsistent system

an augmented matrix corresponds to an inconsistent system only if the last column that is the augmented column is a pivot column

pivot position

an entry that is a pivot of a row echelon form of that matrix

geometric consistency

b is in the span of the columns of A

solution set

collection of all solutions

pivot column

column of a matrix that contains a pivot position

reduced row echelon form

each pivot is one each pivot is the only nonzero entry in its column

Row Reduction Algorithm

every matrix row is equivalent to one and only one matrix in rref

The union of two subspaces of Rn forms another subspace of Rn

false

Given any equation Ax=b, there is a unique x making the equation hold.

false there could be no solution

The equation Ax=b is consistent if the augmented matrix [A b] has a pivot position in every row.

false, the matrix A must have a pivot position in every row

Let A be a matrix with more columns than rows. Is the matrix linearly independent?

false. Only if there are more rows than columns.

In a 5x5 matrix If rankA=4 , then the columns of A form a basis of R5

false. The rank is the dimension of the column space of A by definition and since the rank is 4, the columns of A only span a subspace of dimension 4. But R5 has dimension 5, so the columns A cannot span R5

all tall matrices are one-to-one

false. some tall matrices do not have a pivot in every column.

pivot

first non zero entry of a row in row echelon form

Transformation

if A is an m×n matrix and T(x) = A(x) is the associated matrix transformation: 1) The domain of T is Rn where n is the number of columns 2) the codomain of T is Rm where me is the number of rows 3) The range of T is the column space of A i.e. the number of pivot columns in the transformation matrix A

basis

if B is a basis for a subspace V then any vector in V can be written as a linear combination of vectors in B in exactly one way

when is an equation or system of equations consistent?

if one or more solutions exist

what does asking for a vector equation solution mean?

it is asking if a given vector is a linear combination of some other given vectors

Infinitely many solutions

last augmented column and some other column is not a pivot column

homogeneous system

linear equations where all constant son the right side of equals sign are zero. A homogenous system always has the solution x = 0. This is called the trivial solution. Any nonzero solution is called non trivial

solution

list of numbers x,y,z...that make all equations true simultaneously

Basis of Col(A)

matrix columns corresponding to pivot columns.

1 point) Suppose M and N are 2×2 matrices so that M−N, M+N and M+2N are all non-invertible matrices. Define p(x)=det(M+xN), so that p(x) is a quadratic polynomial, i.e., something of the form ax2+bx+c for some numbers a,b,c. Select the best answer below: p(x)=0 is a quadratic equation with exactly 2 solutions. p(x)=0 is a qauadratic equation, but may have 0, 1 or 2 solutions. p(x)=0 is true no matter what x is. p(x)=0 has exactly 1 solution. There is not enough information in the question to tell.

p(x)=0 is true no matter what x is.

how to find a non zero solution for Ax = 0

plug a non zero value into a free variable

span of the columns of A

set of all b such that Ax=b is consistent. this is always a span. this is a subset of Rm. it is not computed by rref.

span

set of all linear combination of a set of vectors. set of all vectors b in Rn such that x1v1+x2v2+......xkvk = b span is also the subset spanned or generated by vectors v1 to vk

Column Space

span of the columns of a matrix

Basis of a subspace

suppose V is a subspace in Rn. A basis of V is a set of vectors {v1,v2,.....,vm} in V such that: 1) V = Span {v1,v2,.......,vm} 2) the set {v1,v2,.......,vm} is linearly independent A basis is the minimally spanning set if you remove a vector from the set it will no longer span V

Nullity of a matrix A

the dimension of the null space. number of free variables i.e. number of columns without pivots.

Column Space

the subspace of Rm spanned by the columns of A written as Col(A)

free variables come from columns without pivots

true

linear systems of row-equivalent matrices have the same solution set

true

the process of row operations to a matrix does not change the solution set of the corresponding linear equations

true

you can choose any value for the free variables in a consistent linear system

true

row equivalent

two matrices are row equivalent if one can be obtained from the other through row operations

when is an equation or system of equations inconsistent?

when it does not have a solution

inconsistent/empty solution set

when the system is inconsistent the solution set is empty

how to tell when v is in the span of x1 x2 etc

write equation a1x1+a2x2....akxk=v row reduce augmented matrix to determin consistency if the equation is consistent then v is in the span

free variable

xi is a free variable if its corresponding column in A is not a free variable