Exam 3

Question 1

5.1 Eigenvector Definition

Answer 1

a nonzero vector x such that Ax = lambdax for some value of lambda.

Question 2

5.1 eigenvalue

Answer 2

lambda is a eigenvalue of A if there is a nontrivial solution x of Ax = lambdax

Question 3

5.1 Theorem 1

Answer 3

eigenvalues of a triangular matrix are the entries on its main diagonal

Question 4

5.1 Theorem 2

Answer 4

If the eigenvectors correspond to the eigenvalues of a matrix, then the vectors are linearly independent

Question 5

5.2 Invertible Matrix Theorem: A is invertible only if __ is __ an eigenvalue of A

Answer 5

A is invertible only if the number 0 is not an eigenvalue of A

Question 6

What does det AB equal

Answer 6

det A(det B)

Question 7

What does det A^T equal

Answer 7

det A

Question 8

Does row replacement change the determinant

Answer 8

No, but it does change the sign of the determinant

Question 9

How does row scaling effect the determinant

Answer 9

Row scaling scales the determinant by that scalar value

Question 10

multiplicity

Answer 10

how many times an eigenvalue occurs

Question 11

Similarity

Answer 11

A is similar to B if there is an invertible matrix P such that APP^-1 (inverse of P) = B

Question 12

5.3 A matrix is diagonalizable if

Answer 12

A is similar to a diagonal matrix

Question 13

5.3 diagonalization theorem

Answer 13

An matrix is diagonalizable
only if A has enough linearly independent eigenvectors to form a basis of R^n.

Question 14

5.3 A=PDP^-1 (is diagonalizable) only if what

Answer 14

the columns of P are n linearly independent eigenvectors of A

Question 15

5.3 Theorem 6

Answer 15

An nxn matrix with n eigenvalues is diagonalizable

Question 16

5.4 Theorem 8: Diagonal Matrix Representation

Answer 16

If the columns of B form a basis, then D is the basis matrix for the transformation x -> Ax

Question 17

6.1 Inner Product/ Dot Product

Answer 17

[u1, …un] [v1,…vn] = u1v1+… unvn

Question 18

6.1 length/norm of a vector

Answer 18

||v|| = sqrt(v*v) = sqrt(v1^2 + … vn^2)

Question 19

6.1 ||cv|| = ?

Answer 19

|c| ||v||

Question 20

6.1 unit vector

Answer 20

vector whose length is 1

Question 21

6.1 How to find the distance between u and v

Answer 21

dist(u,v) = ||u-v||

Question 22

6.1 Orthogonal

Answer 22

2 vectors are orthogonal if u*v = 0

Question 23

6.1 A vector is perpendicular to W only if what?

Answer 23

x is orthogonal to every vector in a set that spans W.

x * all other vectors = 0

Question 24

6.1Theorem 3

Answer 24

The orthogonal complement of the row space of A is the null space of A. The orthogonal complement of the column space of A is the null space of A^T

Question 25

6.2 Theorem 4

Answer 25

If S is an orthogonal set of nonzero vectors, then S is linearly independent and a basis for the subspace spanned by S

Question 26

6.2 Orthogonal Basis

Answer 26

A basis for W that is also an orthogonal set

Question 27

6.2 Theorem 5: For each y is W the constants in the linear combination y = c1u1…cpup are given by ______

Answer 27

cj = (yuj)/(ujuj)
where j = 1…p

Question 28

6.2 Theorem 6: An mxn matrix U has orthonormal columns only if ____

Answer 28

U^T*U = I(identity matrix)

Question 29

6.2 Theorem 7: Let U be an mxn matrix with orthonormal columns, and let x and y be in R^n. Then ||Ux|| = ?

Answer 29

||x||

Question 30

6.2 Theorem 7: Let U be an mxn matrix with orthonormal columns, and let x and y be in R^n. Then (Ux)*(Uy) = ?

Answer 30

x*y

Question 31

6.2 Theorem 7: Let U be an mxn matrix with orthonormal columns, and let x and y be in R^n. Then
(Ux)*(Uy) = 0 only if ____

Answer 31

x*y= 0

Question 32

6.3 The orthogonal Decomposition Theorem

Answer 32

Let W be a subspace of R^n. Then each y in R^n can be written in the form: y = yhat +z

where yhat is in W and z is perpendicular to W. If {u1…up} is an orthogonal basis of W, then:
Yhat = ((yu1)/u1u1)u1) + …((yup)/upup)up)

and z = y - yhat

Question 33

6.3 If y is in W = Span{u1…up}, then ?

Answer 33

projection of w y = y

Question 34

6.3 The best Approximation Theorem

Answer 34

Let W be a subspace of R^n, let y be a vector in R^n, and let yhat be the orthogonal projection of y onto W.

then yhat is the closest point in W to y.
||y-yhat|| < ||y-v||
for all v in W distinct from yhat

Question 35

6.3 theorem 10. if {u1… up} is an orthonormal basis for a subspace W of R^n then what

Answer 35

projw y = (yu1)u1 + … (yup)up

Question 36

6.3 theorem 10 if U = [u1…up] then what

Answer 36

projw y = UU^T y for all why in R^n

Question 37

4.1 subspace: 3 properties

Answer 37

contains the zero vector
is closed under addition
is closed under multiplication