New Deck

Question 1

Basis (for a subspace)

Answer 1

A basis for a subspace is a set of vectors v_1, …, v_k in W such that :

1. v_1, … v_k are linearly independent; and
2. v_1, … v_k span W

Question 2

Characteristic Polynomial of a matrix:

Answer 2

the characteristic polynomial of a n by n matrix A is the polynomial in t given by the formula det(A-t*I)

Question 3

Column Space of a matrix:

Answer 3

the column space of a matrix is the subspace spanned by the columns of the matrix considered as vectors

Also row space

Question 4

Defective matrix

Answer 4

a matrix A is defective if A has an eigenvalue whose geometric multiplicity is less than its algebraic multiplicity

Question 5

Diagonalizable matrix

Answer 5

A matrix is diagonalizable if its dimension of a subspace: the dimension of a subspace Wi s the number of vectors in any basis of W (if W is the subspace{0}, we say that its dimension is 0)

Question 6

Row echelon form of a matrix

Answer 6

A matrix is in row echelon form if

1. all rows that consist entirely of zeros are grouped together at the bottom of the matrix; and
2. the first counting left to right) nonzero entry in each non zero row appears in a column to the right of the first nonzero entry in the preceeding row (if there is an preceeding row)

Question 7

Reduce row echelon form of a matrix:

Answer 7

A matrix is in reduce row echleoon form if

1. matrix is in row echedlon form
2. the first nonzero entry in each nonzero row is the number 1; and
3. the firs tnonzero entry in each nonzero row is the only nonzero entry in its column

Question 8

Eigenspace of a matrix

Answer 8

The eigenspace associated with the eigenvalue c of a matrix A is the null space of A-c*I

Question 9

Eigenvalue of a matrix:

Answer 9

An eigenvalue of a n by n matrix A is a scalar c such that Ax =cx holds for some nonzero vector x

Question 10

Eigenvector of a matrix:

Answer 10

An eigenvector of a n by n matrix A i a nonzero vector x such that Ax=cx holds for some scalar c

Question 11

equivalent linear systems

Answer 11

Two system of linear equations in n unknowns are equivalent if they have the same set of solutions

Question 12

homogenous linear system

Answer 12

A system of linear equations A*x=b is homogeneous if b=0

Question 13

inconsistent linear system

Answer 13

A system of linear equations is inconsistent if it has no solutions

Question 14

inverse of a matrix

Answer 14

the matrix B is an inverse for the amtrix A if AB = BA = I , identity matrix

Question 15

Least squares solution of a linear system

Answer 15

A leat-squares soution to a system of linear equations Ax = b is a vector x that minimizes the length of the vector Ax-b

Question 16

Linear Combination of vectors

Answer 16

vector v is a linear combination of the vectors v_1, …, v_k if there exist scalars a_1, …, a_k such that v=a_1 * v_1 + …+ a_k*v_k

Question 17

Linearly Dependent vectors

Answer 17

vectors v_1, … , v_k are linearly dependent if the equation a_1 * v_1 + …+ a_k*v_k =0 has a solution where not all the scalar a_1, …, a_k are zero

Question 18

Linearly Independent vectors

Answer 18

vectors v_1, … , v_k are linearly dependent if and only if the solution to the equation a_1 * v_1 + …+ a_k*v_k =0 is the solution where all the scalar a_1, …, a_k are zero

Question 19

Linear Transformation

Answer 19

a linear transformation from V to W is a function T from V to W such that:

1. T(u+v) = T(u) + T(v) fpr all vectors u and v in V; and
2. T(av) = aT(v) for all vectors v in V and all scalars a

Question 20

algebraic multiplicity of an eigenvalue:

Answer 20

The algebraic multiplicity of an eigenvalue c of a matrix A is the number of times the factor (t-c) occurs in the characteristic polynomial of A

Question 21

Geometric mutliplicity of an eigenvalue

Answer 21

The geometric multiplicity of an eigenvalue c of a matrixA is the dimension of the eigenspace of c.

Question 22

Symmetric matrix

Answer 22

a matrix is symmetric if it equals its transpose

Question 23

Subspace

Answer 23

A subspace Q of n-space is a subspace if

1. the zero vector is in W
2. x+y is in W whenever x and y are in W; and
3. a*x is in W whenever x is in W and a is any scalar

Question 24

Span of a set of vectors

Answer 24

the span of the vectors v_1, … v_k is the subset V consisting of all linear combinations pof v_1, … v_k . One can also say that the subspace V is spanned by the vectors v_1, .. , v_k and that these vectors span V

Question 25

Singular matrix

Answer 25

An by n matrix A is singular if the equation A*x = 0 (where x is an n-tuple) has a nonzero solution for x

Question 26

Row space of a matrix

Answer 26

the row space of a matrix is the subspace spanned by the rows of the matrix considered as vectors

Question 27

Row equivalent matrices

Answer 27

Two matrices are row equivalent if one can be obtained from the other by a sequence of elementary row operations:
The elementary row operations performed on a matrix are :
* interchange two rows;
* multiply a row by a nonzero scalar;
add a constant ultiple of one row to another

Question 28

Rank

Answer 28

of a linear transformation

Question 29

Similar Matrices

Answer 29

Matrices A and B are similar if there is a square nonsingular matrix S such that s^{-1} AS=B

Question 30

Nonsingular matrix

Answer 30

an n by n atrix A is nonsingular if the only solution to the equation A*x = 0 is x =0

Question 31

Null Space of a matrix

Answer 31

The null space of a m by n matrix is the ser of all n-tuple x such that A*x=0

Question 32

Null Space of a linear transformation

Answer 32

Null space of a linear transformation T is the set of vectors v in its domain such that T(v) =0

Question 33

Nullity of a matrix

Answer 33

The nullity of a matrix is the dimension of its null space

Question 34

Nullity of a linear transformation

Answer 34

The nullity of a dimension of its null space

Question 35

Orthogonal set of vectors

Answer 35

A set of n-tuples is orthogonal if the dot product of any two of them is 0

Question 36

Orthogonal matrix

Answer 36

A matrix A is orthogonal if A is invertible and its orthogonal linear transformation. A linear transformation T from V to W orthogonal if T(v) has the same length as v for all vectors v in V

Question 37

Orthonormal set of vectors

Answer 37

A set of n-tuples os orthonormal if it is orthonormal if it is orthogonal and each vector has length 1

Question 38

Range of a matrix

Answer 38

The range of a m by n matrix A is the set of all m-tuples A*x, where x is any n-tuples

Question 39

Range of a linear transformation

Answer 39

The range of a linear transformation T is the set of all vectors T(v) , where v is any vector in tis domain

Question 40

Rank of a matrix

Answer 40

The rank of a matrix is the number of nonzero rows in any row equivalent matrix that is in row echelon form

Question 41

Rank of a linear transformation

Answer 41

the rank of a linear transformation (and hence of any matrix regarded as a linear transformation )is the dimension of its range. A theorem tell us that the two definitions of rank of a matrix are equivalent

Question 42

row equivalent matrices

Answer 42

two matrices are row equivalent if one can be obtained from the other by a sequence of elementary row operations:
The elementary row operations performed on a matrix are:
interchange two rows;
multiplying a row by a nonzero scalr
add a constant multiple of one row to another

Question 43

Sheldon Axler Notes:

Vectors are defined first

Linear independence, span , basis and dimension are defined in the chapter , which present the basic theory of finite-dimensional vector spaces

Linear maps are introduced next leading to Fundamental Theorem of Linear Maps: if T is a linear map on V, then dimV - dim nullT = dim range T. Quotient spaces and duality are topics with high abstraction

Theory of polynomials is needed to understand linear operators (no linear algebra).

Studying linear operators by restricting it to small subspaces leads to eigenvectors. Complex vector spaces, eigvenvalues always exist. Eachlinear operator on a complex vector space has a upper triangular matrix with respect to some basis.

Inner product spaces are defined in this chapter and their basic properties are developed along with orthonormal bases and Gram-schimdt procedure. Orthogonal projections can be used to solve certain minimalization problems

Spectral theorem characterizes linear operators where there is an orthonormal basis consisting of eigenvector is the highlight.

Answer 43

notes

Question 44

Invertible Matrix

Answer 44

How do you know if A^{-1} exists: no nonzero solution to Ax = 0

columns are linearly independent of A

rows of A are Linearly Independent

detA != 0

unique solution to Ax=B

all eigenvalues are nonzero: means it is invertible
nonsingular means invertible

Question 45

if Ax=0, only solution is 0 if A is invertible and ths A^{-1} A = I

Answer 45

How do you now if you can invert A, IF A is invertible A must be nonsingular (A^{-1} is also nonsingular)

Det(A) != 0 same as Ax=0 same as x=0

Question 46

Solutions to system

Consistent: IFF right most column is not

Inconsistent:

Answer 46

consistent and inconsistent

Question 47

Reduced Row Echelon Form (RREF)

Answer 47

Free variable if pivot has 0 below it

unique solution : no free variable

infinitely many solutions: at least 1 free variable

Question 48

Linear combination

Answer 48

y = c_1x_1 + …. + c_nx_n

Question 49

set

Answer 49

{[ ] , [ ] , [ ] }

Question 50

Span

Answer 50

some linear combination of set

Question 51

identity matrix

Answer 51

1 0 0
0 1 0
0 0 1