Final Exam

Question 1

equation of line through p parallel to v

Answer 1

x = p + tv

Question 2

domain of T (based on an m × n matrix)

Answer 2

Rⁿ

Question 3

codomain of T (based on an m × n matrix)

Answer 3

R^m

Question 4

range of T

Answer 4

all images T(x)

Question 5

linear transformation of a m × n matrix

Answer 5

• Rⁿ → R^m
• T(u + v) = T(u) + T(v)
• T(cu) = cT(u)

Question 6

standard matrix of a linear transformation T

Answer 6

• T(x) = x₁T(e₁) + x₂T(e₂) + … + x_n(Te_n)
  *

Question 7

one-to-one and onto

Answer 7

• one-to-one: mapping T: Rⁿ → R^m if each b in R^m is the image of at most one x in Rn (T(x) = 0 has only the trivial solution)
• onto: mapping T: Rⁿ → R^m if each b in R^m is the image of at least one x in Rⁿ

Question 8

invertible matrix

Answer 8

for a square matrix A, another matrix C exists for which AC = CA = I

Question 9

properties of invertible matrices

Answer 9

• x = A^-1b
• (AB)^-1 = B^-1A^-1
  *

Question 10

alogrithm for finding A^-1

Answer 10

row reduce the augmented matrix [A | I]

Question 11

Invertible Matrix Theorem

Answer 11

1. A is invertible
2. A is row equivalent to the n × n identity matrux
3. A has n pivot positions
4. Ax = 0 has only the trivial solution
5. the columns of A form a linearly independent set
6. x → Ax maps Rⁿ to Rⁿ
7. Ax = b has at least one solution for each b in Rⁿ
8. x → Ax is one-to-one
9. the columns of A span Rⁿ
10. there exists a n × n matrix C such that CA = I
11. there exists a n × n matrix D such that AD = I
12. A^T is inverstible

Question 12

properties of determinants

Answer 12

• elementary row operations
  
  • roe replacement: detB = detA
  • interchange: detB = -detA
  • scalar multiplication: detB = kdetA
• a square matrix is invertible if detA does not equal 0
• for n × n matrices, detA^T = detA
• detAB = (detA)(detB)

Question 13

Cramer’s Rule

Answer 13

• for any b in Rⁿ, the unique solution x of Ax = b has entries given by x_i = detA_i(b) ÷ detA
• replace a column with b, find the determinant of the new matrix, and divided by the determinant of the original matrix to find the solution for the variable corresponding to the column replaces

Question 14

formula for A^-1

Answer 14

A^-1 = adjA ÷ detA, where adjA = transpose of the matrix of cofactors

Question 15

calculating area/volume using matrices

Answer 15

• for a 2 × 2 matrix, the area of the parallelogram detrmined by the columes of A is |detA|
• for a 3 × 3 matrix, the volume of the parallelpiped determined by the columns of A is |detA|
• one point must be at the vertex

Question 16

subspace of vector space V

Answer 16

• subset of vector space V
  
  • zero vector of V contained in H (H contains {0})
  • H closed under vector addition (for every u and v in H, u + v also in H)
  • H closed under multiplication by scalars (for each u in H, cu also in H)

Question 17

nulA

Answer 17

• set of all solutions of the homogenous equation Ax = 0
• subspace of Rⁿ
• span of the vector equation representation of the solution set of Ax = 0

Question 18

colA

Answer 18

• set of all linear combinations of the columns of A
• subspace of R^m

Question 19

linear transformations of vector spaces (and subspaces)

Answer 19

• linear transformation: occurs from a vector space V to a vector space W, assigning each vector x in V a unique vector T(x) in W which ovey all laws of vector spaces (ie, closed under addition and multiplication by scalars)

Question 20

kernel and range

Answer 20

• kernel: nulT (set of all u in V such that T(u) = 0 in W)
• range: set of all vectors in W of the form T(x) for some x in V

Question 21

linear independence in vectors

Answer 21

• an indexed set of vectors (eg, {v₁, …, v_p}) in which v₁ does not equal zero is linearly dependent if and only if some v_j with j > 1 is a linear combination of the preceding vectors

Question 22

basis

Answer 22

• B ({b₁ … b_p}) is a basis for the subspace H if
  
  • B is linearly independent
  • H = span{b₁ … b_p} (ie, subspace spanned by B coincides with H)

Question 23

bases for nulA and colA

Answer 23

• nulA = set of solutions to the homogenous equation
• colA = pivot columns of A (row-reduce to determine)

Question 24

change-of-coordinates matrix

Answer 24

• x = P_B[x}_B
• P_B^-1x = [x]_B

Question 25

rowA

Answer 25

set of all linear combinations of the row vectors in A

Question 26

rank

Answer 26

• dimension of colA
• rankA + dimnulA = n

Question 27

change-of-basis matrix

Answer 27

• for bases B = [b1 … bp] and C = [c1 … cp] in vector space V, there exists a unique square matrix such that [x}_C = P_B→C[x]_B
• to find P_B→C: [c₁ … c_n | b₁ … b_n] ~ [I | P_B→C]

Question 28

eigenvectors and eigenvalues

Answer 28

• eigenvector - nonzero vector x for a n × n matrix A such that Ax = λx for some scalar λ
• eigenvalue - scalar λ for which a nontrivial solution x exists such that Ax = λx

Question 29

determining eigenvectors/eigenvalues for a matrix A

Answer 29

• potential eigenvectors given: multiply matrix A by potential eigenvector; if the result is a scalar multiple of the potential eigenvector, then it is an eigenvector of A
• potential eigenvalue given: (A - λI) = 0

Question 30

basis for eigenspaces of A

Answer 30

1. compute (A - λI)
2. row reduce the augmented matrix for (A - λI) = 0
3. rewrite as a vector-equation, if necessary

Question 31

distinct eigenvalues theorem

Answer 31

if v₁ … v_p are eigenvectors that crrespend to distinct eigenvalues λ₁ … λ_r of a n n matrix A, then the set {v₁ … v_p} is linearly independent

Question 32

the characteristic equation

Answer 32

• det(A - λI)
• a scalar λ is an eigenvalue of a n × n matrix A if and only if λ satisfies the characteristic equation

Question 33

similar matrices

Answer 33

• an invertible matrix p exists such that A = PBP^-1, or B = P^-1AP
• similar matrices share a characteristic polynomial and hence eigenvalues (including multiplicities)

Question 34

criteria for diagonalization

Answer 34

a n × n matrix A is diagonalizeable if and only if A has n linearly independent eigenvectors

Question 35

diagonalization

Answer 35

1. find the eigenvalues of A
2. find n linearly independent eigenvectors of A
3. construct P using the linearly independent eigenvectors
4. construct D from the corresponding eigenvalues

Question 36

u _º v

Answer 36

u₁v₁ + u₂v₂ + … + u_pv_p

Question 37

length of v (||v||)

Answer 37

√(v _º v) = √(v₁v₁ + v₂v₂ + … +v_pv_p)

Question 38

||cv||

Answer 38

|c| × ||v||

Question 39

distance between u and v (dist(u,v))

Answer 39

||u - v|| = √(u₁ - v₁)² + (u₂ - v₂)² + … (u_p - v_p)²

Question 40

orthogonal vectors

Answer 40

• dist(u, -v) = ||u||² + ||v||² + 2u_ºv
• u_ºv = 0

Question 42

orthogonal compliments

Answer 42

• x is an orthogonal compliment to W if and only if x is orthogonal to every vector in a set that spans W^┴
• W^┴ is a subspace of Rⁿ
• (RowA)^┴ = NulA
• (ColA)^┴ = NulA^T

Question 43

find a unit vector u in the same direction as v

Answer 43

• compute ||v||
• multiply v by (1 ÷ ||v||)
• check that ||u|| = 1

Question 44

orthogonal set

Answer 44

if S = {u₁, … u_p} contains nonzero vectors in Rⁿ all orthogonal to one another

Question 46

properties of orthogonal sets

Answer 46

• if S = {u₁, …, u_p}, an orthogonal set of nonzero vectors in Rn, then S is linearly independent and hence is a basis for the subspace spanned by S

Question 47

orthogonal basis

Answer 47

c_j = (y _º u_j) ÷ (u_{j º} u_j)

Question 48

orthogonal projection of y onto L

Answer 48

proj_Ly = u[(y _º u) ÷ (u _º u)]

Question 49

distance from y to L

Answer 49

||y - (proj_Ly)||

Question 50

orthonormal sets and orthonormal bases

Answer 50

• orthonormal set: orthogonal set of unit vectors
• orthonormal basis: orthonormal set which spans a subspace W

Question 51

express