Math 319 Final

Find A-λI and then the det(A-λI) by multiplying the crosses (you'll get something like (x-λ)(x-λ)-(value)). Simplify and factor to find the two eigenvalues.

Find the eigenvalues of the matrix A =[2x2]. Show your work

Set A equal to [1,0,0;0,1,0;0,0,1] and row reduce until [1,0,0;0,1,0;0,0,1] is on the left side. The right side [3x3] of the matrix is A^-1.

Find the inverse of the matrix.A = [3x3 composed of 0's and 1's].

a) TRUE b) FALSE c) FALSE d) TRUE

For each of the following statements, circle TRUE or FALSE. No justification is required. (a) It is possible for the columns of a 3x4 matrix to span R3. (b) It is possible for the columns of a 4x 3 matrix to span R4. (c) It is possible for the columns of a 3x 4 matrix to be linearly independent (d) It is possible for the columns of a 4x3 matrix to be linearly dependent.

rows

The column space ColA is a subspace of ℝ^k where k is the number of ______ in A

column

The columns of a matrix A are linearly independent if and only if there is a pivot in every _______ of A

every vector b in ℝ^m is a linear combination of the columns of A

The columns of an mxn matrix span ℝ^m if...

a) # pivot columns b) # free variables = # non-pivot columns c) dim(ColA) = # pivot columns

The matrix A and an echelon form of A are given below. A = [4x5] ~ [reduced 4x5] a) dim(ColA) b) dim(NulA) c)Rank(A)

a) the pivot columns of A are the basis for Col(A) (find the pivots in the reduced and the same columns in the original are the vectors in the basis) b) Find the parametric vector form from the reduced echelon form. The resulting vector is the basis for Nul(A) c) Rank(A) = dim(ColA). Nullity(A) = dim(NulA)

The matrix A and the reduced echelon form of A are given below. A = [4x4] ~ [reduced 4x4] (a) Find a basis for Col(A) (b) Find a basis for Nul(A) (c) Find the rank and nullity of A.

row

The matrix equation Ax = b is consistent for all b if and only if there is a pivot in every ____ of A

columns; rows

The matrix product AB is defined only when the number of ______ of A matches the number of ____ of B

the number of times (λ-a) occurs as a factor of the characteristic polynomial

The multiplicity of an eigenvalue λ=a is...

columns

The null space Nul A is a subspace of ℝ^k where k is the number of _____ of A

columns

The rank of A plus the dimension of Nul A is equal to the number of ______ of A

all linear combinations of v1,v2,...,vp

The set Span{v1,v2,...,vp} consists of...

1) contains zero vector 2) closed under addition 3) closed under scalar multiplication

What three properties must hold in order for a subset H of ℝ^n to be a subspace of ℝ^n?

Create a matrix with all the vectors and set it equal to b. Reduce to reduced echelon form and find the values for x1,x2,x3. Then write those values as weights for a1,a2,a3 in the linear combination (b = xa1 + xa2 + xa3)

Write b as a linear combination of a1,a2, and a3, or explain why this is not possible. a1 = [1x3], a2 = [1x3], a3 = [1x3], b = [1x3].

a system of linear equations with at least one solution

a consistent system is...

a) rank(A)+nullity(A)=n b) solve for Nullity(B) by using the equation

a) Fill in the blanks to complete the Rank-Nullity Theorem: If A is an mxn matrix, then ____+_____=___ b) Let B be a 6x8 matrix and suppose Rank(B) = 6. What is Nullity(B)? Explain your answer.

a) 1/(ab-cd)*[d,-b;-c,a] b) Multiply A^-1 by vector of x and y.

a) Find the inverse of A = [a,b;c,d] b) Use your answer to solve the system a + b = x c + d = y

for every b in ℝ^m the equation T(x) = b has at most one solution

A linear transformation T: ℝ^n -> ℝ^m is one-to-one if...

for every b in ℝ^m the equation T(x) = b has at least one solution

A linear transformation T: ℝ^n -> ℝ^m is onto if...

A is similar to a diagonal matrix. That is, if there exists invertible matrix P + diagonal matrix D such that A = PCP^-1

A nxn matrix A is diagonalizable if...

there are weights c1, c2,..., cp not all zero such that c1v1+c2v2+...+cpvp = 0 (zero vector)

A set of vectors {v1,v2,...,vp} is linearly dependent if...

the only solution to x1v1+x2v2+...+xpvp = 0 is the trivial solution x1 = x2 = ... = 0 (vector zero)

A set of vectors {v1,v2,...,vp} is linearly independent if...

c1v1+c2v2+...+cpvp = y with weights or scalars c1, c2,..., cp

A vector y is a linear combination of the vectors v1, v2,...,vp if...

1) linearly independent 2) spans ℝ^n

According to the def, what two properties must hold in order for a set of vectors {v1,v2,...,vp} in ℝ^n to be a basis for ℝ^n?

a nonzero vector x such that Ax = λx for some scalar λ

An eigenvector of a matrix A is...

there exists a matrix C satisfying AC = In and CA = In

An nxn matrix A is invertible if...

Reduce. Echelon form has zeros under each pivot. Put it in x equals form and solve for the free vars. should be in x1[]+xp[] form.

Below is the augmented matrix of a linear system Ax = b. Reduce the matrix to reduced echelon form to find the general solution to the system. Show your work, and circle the first time in your work that the matrix achieves echelon form (not reduced)

The equation det(A-λI)=0 of A

Characteristic equation

Find the row with the least amount of nonzero values. Remember the +, - pattern and cross out row+column nonzero is on and write the form out (ex. -0+0-3|(4x4)| +0-0). Keep doing this until you get 2x2. Cross multiply left to right to get values (topL*bottomR + topR*bottomL). Then simplify to get a single num.

Compute the determinant using cofactor expansions. |(5x5)|

a) Just combine vectors into a matrix b) Multiply that matrix by the 2x1 matrix c) Set v equal to the matrix and solve for x1 and x2

Consider the following basis for R^2: B =([2x1],[2x1]) a) Find the change of coordinates matrix PB that transforms B-coordinates into standard coordinates b) Suppose [x]B = [2x1]. Write x in standard coordinates. c) Let v = [2x1] (in standard coordinates). Find [v]B

use row of operations to make A look like B. If you switch rows, multiply det(A) by -1. If you divide a row by a value, multiply det(A) by the denominator of that fraction. Resulting det is det(B).

Consider the matrices A = [3x3] (has a, b, and c values) , B = [3x3] (has a, b, c values and signs +,-) .If det(A) = x (value), what is det(B)? Show your work and justify your answer.

a) Multiply matrices b) Not defined # columns of A does not equal # rows of B c) Change B to [2x3] and multiply C by 2. Then add together.

Consider the matrices A=[2x2], B=[3x2] , C=[2x3]. Compute the given matrix operation, or explain why it is not defined. a) BA b) AB c) B^T + 2C

a) C and D are same size so you can subtract b) B and C are NOT the same size so not defined c) # of cols of A do NOT equal # rows of C so not defined d) # of cols of A = # of rows of B so yes, multiply. e) # of cols of B = # of rows of A so yes, multiply.

Consider the matrices. A = 2x3 B = 2x3 C= 2x2 D= 2x2 a) 3C-D b) B+C c) AC d) AB e) BA

a) Multiply A with the 3x1 matrix. If the zero vector is not the result then no. A[3x1] does not equal 0 so not in Nul(A). b) Any linear combination of the columns of A if they were vectors. (ex. v1+v2 or v2+v3)

Consider the matrix A, given by A = [3x3]. a) Is [3x1] in Nul(A)? b) List two non-zero vector in Col(A) which are not columns of A

a) Combine the vectors into a matrix and reduce to echelon form. The pivot columns of the original matrix make up the basis for H. b) The dimension is the number of vectors in the basis c) Combine the vectors in the basis and set [] equal to it. Reduce to matrix to echelon form and if it is consistent = yes, inconsistent = no.

Consider the subspace H = Span{vectors}. (a) Find a basis for H. Show your work and justify your answer. (b) Recall the subspace H from the previous page. What is the dimension of H? Justify your answer. (c) Is [] in H? Explain your answer.

a) Use the zero vector, v1, v2, v1+v2, v1-v2 b) Span {v1,v2} is the plan in R^3 that contains v1,v2, and 0

Consider the vectors v1=[1x3], v2=[1x3]. a) List 5 vectors in Span{v1,v2}. b) Give a geometric description of Span{v1,v2}

there is an invertible matrix P with B = P^-1AP

Considering two nxn matrices A and B, we say A is similar to B if...

Combine vectors into a matrix and set equal to zero. Reduce to echelon form and solve for x. The value of each x becomes its weight in the linear dependence relation (x1v1+x2v2+x3v3 = 0)

Demonstrate that the set of vectors {v1, v2, v3} is linearly dependent by finding a linear dependence relation. Show your work. v1 = [1x2], v2 = [1x2], v3 = [1x2]

Set A equal to b and reduce until you get to echelon form. should get [0 0 (equation)]. Consistent if (equation), then simplify to get b2 = or b1 =.

Describe the set of all b for which Ax = b is consistent. A = [2x2], b = [b1;b2]. Ax = b is a consistent system when...

(i) Dependent (ii) Dependent (iii) Independent (iv) Dependent

Determine by inspection whether each of the following sets of vectors are linearly inde-pendent or linearly dependent. Justify your answer in a few words. (Exam 1 Spring 2020 Page 9)

(a) SOMETIMES (could be inconsistent) (b) NEVER (too many variables... one must be free) (c) SOMETIMES (d) NEVER (a system can have 0,1, or infinitely many solutions) (e) ALWAYS (must have a pivot at first slot)

Determine if each of the following statements is ALWAYS true, SOMETIMES true, or NEVER true (a) A system of 3 equations and 4 unknowns has infinitely many solutions (b) A system of 3 equations and 4 unknowns has a unique solution (c) A system of 4 equations and 3 unknowns is consistent (d) A system of 4 equations and 3 unknowns has exactly 3 solutions (e) A system of 1 equation and 3 unknowns has infinitely many solutions

Create a matrix with the vectors and reduce to echelon form. It is a basis if there is a pivot in every row and column because that means it spans R^3 and is linearly independent

Determine whether the following set of vectors is a basis for R^3. Justify your answer. {[set of three vectors]}

Combine the vectors into a matrix and reduce to echelon form. If there is a pivot in every column = linearly independent. If there isn't = linearly dependent

Determine whether the set of vectors {v1, v2, v3} is linearly independent or linearly dependent. Show your work and explain your answer. v1 = [1x4], v2 = [1x4], v3 = [1x4].

a) find det(A-λI) by getting the diagonal -λ. Then pick a row/column to use to create a formula using the crossing out method. (dont forget about +/-) Then simplify until you can factor out. You may get multiples. b) Using the eigenvalues from A, use them as the λ values for A-λI then reduce to reduced echelon form. Put the answer in parametric vector form and the basis is the vectors resulting from that form. c) Combine the vectors from the basis's to get P. Set diagonal values of a 3x3 matrix to the eigenvalues to get D.

Diagonalize A = [3x3] a) Find the eigenvalues of A b) Find a basis for each eigenspace of A c) Find an invertible matrix P and a diagonal matrix D such that A = PDP^-1

Reduce to echelon form. Is there a pivot in every row? If so, the columns span R^4

Do the columns of A span R^4? Show your work and explain your conclusion. A = [4x4]

A scalar λ of A corresponding to the eigenvector x

Eigenvalue

a) No, zero vector does not satisfy x3 = x1+ (nonzero) because 0 does not equal 0 + (nonzero). So zero vector is not in G. b) Yes (Page 6 of Exam3A)

For each set G below, determine whether G is a subspace of R^3. Justify your answer: If G is not a subspace, explain which property fails. If G is a subspace, verify that the three subspace properties hold. a) G = {[x1;x2;x3] : x1 + (nonzero)} Is G a subspace? b) G = {[x1;x2;x3] : x1 + x2} Is G a subspace?

a) G is not closed under addition => not a subspace b) G is not closed under scalar multiplication => not a subspace

For each set G below, give a counterexample (two specific vectors or a specific vector and scalar) to demonstrate that G is not a subspace of R^3. a) G = {[x1,x2,x3]: x1x2=x3} b) G = {[x1,x2,x3]: x1= |x3|}

Combine vectors into a matrix and set the h vector equal to it. Reduce to echelon form and you should have an h equation. The system has to be consistent for the vector to be in the Span so make sure the h value allows this. (ex. if h-2 then h needs to = 2)

For which value(s) h is the vector [1;1;h] in Span {[two vectors 3x1]}?

Row reduce |(comb of letters)| to get it back to the simplest form. If you swap rows, multiply x by -1. If you divide a row to simplify, multiply x by denominator. Resulting value is the determinant.

If |a,b,c;d,e,f;g,h,i| = x, find the determinant of the matrix below. |(comb of letters)|

1) A is invertible 2) A is row equivalent to In 3) A has n pivot positions 4) Ax = 0 has only the trivial solution 5) Columns of A are linearly independent 6) T(x) = Ax is one-to-one 7) Ax = b has a solution for every b in R^n 8) Columns of A span R^n 9) T(x) = Ax is onto 10) A^T is invertible 11) Columns of A form a basis for R^n 12) Col(A) = R^n 13) Rank(A) = n 14) Nullity(A) = 0 15) Nul(A) = {0} 16) det(A) is not equal to 0 17) λ = 0 is not an eigenvalue of A

Invertible Matrix Theorem

a) Set A equal to zero vector and reduce to reduced echelon form. find the vector x = form (ex. x= x2[]) b) Pick a column in A (or any scalar mult of that)

Let A = [4x2] List one non-zero vector in Nul(A) and one non-zero vector in Col(A). a) Vector in Nul(A): b) Vector in Col(A):

a) multiply x and y => xy b) Since A is 2x2 multiply det(A) by 4 c) you can't add determinants (not enough info) d) 1/det(B)

Let A and B be 2x 2 matrices with det(A) = x and det(B) = y. Compute the following determinants. If the determinant cannot be computed using the information given, state "not enough information" a) det(AB) b) det(2A) c) det(A+B) d) det(B^-1)

a, d, e, g (The other three are false because matrix multiplication is not commutative)

Let A and B be invertible nx n matrices. Which of the following statements must be true? Select all that apply. a) (A^1)T = (A^T)^1 b) AB = BA c) ABA^-1 = B d) A^-1AB = B e) (AB)^T = B^TA^T f) (AB)^-1 = A^-1B^-1 g) (AB)^-1 = B^-1A^-1

a, b (A has fewer than n pivots, so all statements in IMT are false)

Let A be a 3x3 matrix and suppose that A has only 2 pivot positions. Which of the following statements must be true about A? Select all that apply. a) There is a vector b in R3 such that Ax = b is inconsistent. b) The equation Ax = 0 has infinitely many solutions. c) The transformation T (x) = Ax is onto. d) A is an invertible matrix. e) The columns of A are linearly independent

a) and c)

Let A be a 4x4 matrix and suppose two of the columns of A are identical.Which of the following statements must be true about A? Select all that apply. a) The columns of A are linearly dependent. b) A is an invertible matrix. c) The equation Ax = 0 has infinitely many solutions. d) The transformation T (x) = Ax is onto.

a, b, d. (d because col don't span R^5)

Let A be a 5 ⇥ 5 matrix and suppose one of the columns of A is the zero vector (i.e. A has a column of all 0's). Which of the following statements must be true? Select all that apply. a) The columns of A are linearly dependent. b) The equation Ax = 0 has infinitely many solutions. c) The columns of A span R5. d) There is a vector b in R5 such that Ax = b is inconsistent. e) A has exactly 4 pivot positions. f) A is an invertible matrix.

No. If A is 5x7, there can be at most 5 pivot positions. So Rank(A) <= 5. If Nullity(A)=1 then we would have Rank(A)+1=7 => Rank(A)=6. not possible

Let A be a 5x7 matrix. Is it possible for the dimension of Nul(A) to equal 1? Explain your answer.

f) We do not have enough information to determine whether the system is consistent Explanation: Since A has 4 rows and only 3 pivots it is possible for the augmented matrix [A | b] to have a pivot in the b column, making Ax=b inconsistent for those b. It's also possible that [A | b] has only 3 pivots, in which case Ax=b is consistent

Let A be a matrix with 4 rows and 4 columns, and suppose A has 3 pivot positions. Let b be a vector in R^4. Which of the following best describes the solution set of Ax = b? a) The system is consistent, and there is exactly one solution. b) The system is consistent, and there are exactly two solutions. c) The system is consistent, and there are infinitely many solutions. d) The system is consistent, but we do not have enough information to determine whether there is one or infinitely many solutions. e) The system is inconsistent. f) We do not have enough information to determine whether the system is consistent

c) The system is consistent, and there are infinitely many solutions. Explanation: There is a pivot in every row of the coefficient matrix A, so the system will be consistent for every b. A has 6 columns and only 4 pivots so there will be 2 free variables. So Ax=b has infinitely many solutions

Let A be a matrix with 4 rows and 6 columns, and suppose A has 4 pivot positions. Let b be a vector in R^4. Which of the following best describes the solution set of Ax = b? a) The system is consistent, and there is exactly one solution. b) The system is consistent, and there are exactly two solutions. c) The system is consistent, and there are infinitely many solutions. d) The system is consistent, but we do not have enough information to determine whether there is one or infinitely many solutions. e) The system is inconsistent. f) We do not have enough information to determine whether the system is consistent

a) The system is consistent, and there is exactly one solution Explanation: All homogeneous equations (Ax = 0) have at least one solution (trivial). This system cannot have infinitely many solutions because it has a pivot in every column and therefore no free variables

Let A be a matrix with 7 rows and 3 columns, and suppose A has 3 pivot positions. Which of the following best describes the solution set of Ax=0? a) The system is consistent, and there is exactly one solution b) The system is consistent, and there are exactly four solutions c) The system is consistent, and there are infinitely many solutions d) The system is consistent, but we do not have enough info to determine whether there is one or infinitely many solutions e) The system is inconsistent f) We do not have enough info to determine whether the system is consistent

(c) The columns of A are linearly dependent (The others are true by the IMT)

Let A be an mxn matrix, and suppose A has a pivot position in every row. Which of the following is not necessarily true? a) For each b in Rm, the equation Ax = b is consistent. b) Each b in Rm is a linear combination of the columns of A. c) The columns of A are linearly dependent. d) The columns of A span Rm. e) None of the above; all of these statements are true

The set of all linear combinations of columns of A

Let A be an mxn matrix. What is the definition of the column space Col(A)?

The set of all solutions to Ax = 0 (zero vector)

Let A be an mxn matrix. What is the definition of the null space Nul(A)?

A,B,C,D,F,H,I

Let A be an n ⇥ n invertible matrix. Which of the following must be true?Select all that apply. (You do not need to show work or give any justification). (A) The columns of A form a basis for R^n. (B) det(A) 6 = 0. (C) The equation Ax = 0 has only the trivial solution. (D) The rank of A is n. (E) The columns of A are linearly dependent (F) The null space NulA consists of exactly one vector. (G) A is diagonalizable. (H) The rows of A form a basis for R^n. (I) λ = 0 is not an eigenvalue of A. (J) The column space ColA consists of exactly n vectors.

a) Yes because ABC = ACD => A^-1(ABC) = A^-1(ACD) => BC = CD b) No because matrix mult is not commutative. We can say (BC)C^-1 = (CD)C^-1 but that only gives us B = CDC^-1. We cannot rearrange.

Let A, B, C, and D be 3x3 invertible matrices and suppose that ABC = ACD. (a) Do we know that BC = CD? Explain. (b) Do we know that B = D? Explain.

No. By invertible matrix theorem, if B is invertible, then B^T is invertible. Also by IMT, if B^T is invertible, then the columns of B^T must be linearly independent.

Let B be a 6 ⇥ 6 invertible matrix, and let BT be the transpose of B. Is itpossible for the columns of BT to be linearly dependent? Explain why or why not

a) Find the vectors a, b, c (ex. a[1;0;0;-1]+ b[...]+c[...]). The vectors are the spanning set. b) Create a matrix with the vectors of a,b,c. Reduce to echelon form. Pivot columns are the basis for H. c) # of vectors in basis

Let H = {[comb of letters] : a, b, c in R}. a) Find a spanning set for H. b) Find a basis for H. c) What is the dimension of H?

The number of vectors in a basis for H

Let H be a subspace of ℝ^n. What is the definition of the dimension of H?

a) Find the combination of the two T matrices that creates T([1x2]) and the resulting vector is the answer. b) The standard matrix is 2x2 [1,0;0,1]. Combine the two T matrices that create the standard matrix

Let T : R2 -> R3 be a linear transformation such that T([1x2]) = [1x3] , T([1x2]) = [1x3]. (a) What is T([1x2])? Justify your answer (b) Find the standard matrix of T . Justify your answer.

a&b) Find the combination of the two T's that gives you the vector T c) You only get the zero vector

Let T : R3 -> R3 be a linear transformation, and suppose that T([3x1])=[3x1] and T([3x1])=[3x1] Compute the following: a) T([3x1]) b) T([3x1]) c) T([0;0;0])

a) Multiply A and u. Resulting vector is T(u) b) Set A equal to zero vector and reduce until you can find the x values. (vector x = []) c) Set A equal to vector b and if it has a solution (Ax=b) then yes it is in the range (T(x)=b has a solution). If not, then no. d) If the only solution to T(x)=0 is the trivial solution (pivot in every column of A) then yes. e) If the vector b from c was in the range of T then yes. if not, then no.

Let T : R^2 -> R^3 be the linear transformation defined by T (x) = Ax, where A = [3x2] a) Compute T (u) where u = [2x1] b) Find all vectors x in R2 such that T (x) = [3x1 0's] c) Is the vector b = [3x1] in the range of T ? Explain. d) Is the transformation T one-to-one? Explain. e) Is the transformation T onto? Explain

a) Multiply A and u. The resulting vector is T(u) b) Set A equal to T(x) and solve for x. Put it in x= form, pick a value for the free var, and the resulting vector is T(x). c) If A has a pivot in every row = yes. if not = no. d) If A has a pivot in every column = yes. if not = no

Let T : R^3 -> R^2 be the linear transformation defined by T (x) = Ax, where A = [2x3]. a) Compute T(u) where u = [3x1] b) Find a vector x in R3 such that T (x) = [2x1], or explain why no such vector exists. c) Is the transformation T onto? Explain. d) Is the transformation T onto? Explain.

Organize the matrix so the first column is a, second is b, etc.

Let T : R^4 -> R^3 be the linear transformation given by the rule T([a; b; c; d]) = [random combo of a, b, c, d]

Set A equal to B and reduce to echelon form. a) whatever values of h and k leaves a pivot in the last column b) whatever values of h and k that makes sure there are no free vars c) whatever values of h and k leaves a free var (a row of all zeros)

LetA = [], b = [] .Find the value(s) of h and k for which the system Ax = b has.. a) No solution b) Exactly one solution c) Infinitely many solutions

rank(A) + nullity(A) = n (number of columns in A)

Rank-Nullity Theorem

a&b) Find A-λI and then reduce the matrix to reduced echelon form and find the x= form (x[]+x[]). The eigenspace will be the set of these vectors ({[],[]}. c) The invertible matrix P is the combination of the two basis's. The diagonal matrix D is the λ values in a diagonal based on how many vectors are in the basis

Remember to show all work. In this problem we will diagonalize the matrix A = [3x3] .The eigenvalues of A are λ = -8 and λ = 2. a) Find a basis for the eigenspace of A corresponding to λ = -8 b) Find a basis for the eigenspace of A corresponding to λ = 2 c) Find an invertible matrix P and a diagonal matrix D such that A = PDP^-1

a system of linear equations with no solution

an inconsistent system is...

the set of all solutions to (A-λI)x = 0 (zero vector)

eigenspace of A corresponding to λ is...

1) consists of all eigenvectors corresponding to λ and the zero vector 2) is a subspace of ℝ^n (it is Nul(A-λI)

the two properties of an eigenspace are...