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Linear Algebra > Final exam > Flashcards

Final exam Flashcards

1
Q

Statements equivalent to A is invertible

A

A^T is invertible, rows are linearly independent, rows span R^n, columns are linearly independent, columns span R^n, rows are equivalent to the identity matrix, A can be written as the product of elementary matrices, determinant can’t be 0, eigenvalue cant be 0, Ax=b only one solution, the null space is the zero vector

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2
Q

Subspace definition

A

A subset of R^n that satisfies two categories: of x,y are elements of U, then their sum is an element of U
Alpha(x) is an element of U

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3
Q

Column space

A

Span of the linearly independent columns of A

Defined by Ax

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4
Q

Row space

A

Span of linearly independent rows

A^Tx

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5
Q

Null space

A

Space where Ax=0

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6
Q

Col(AB) is in or equal to the col(A)

A

This works because taking a vector y in AB means that ABx=y, which you can say is Az=y for Bx=z, therefore y is also an element of col(A)

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7
Q

Row(AB) is in or equal to row(A)

A

Same idea, just use transposes

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8
Q

Rank theorem

A

Rank of matrix+ nullity= number of columns

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9
Q

Linear transformation definition

A

T(x+y)= T(x)+T(y) and T(ax)=aT(x)

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10
Q

Matrices and linear transformations

A

There’s always a matrix of transformation for linear trans such that T(x)=Ax

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11
Q

Coordinate vector

A

Coefficients of linear combo of vectors

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12
Q

Eigenvalues

A

There’s an eigenvalue for square matrix such that Ax=(lambda)x

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13
Q

Geometric multiplicity

A

Dimension of the eigenspace

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14
Q

Trace

A

Sum of the diagonal is a matrix

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15
Q

Eigenvalues with transposes

A

A and A^T have the same eigenvalues

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16
Q

Properties of determinants

A

If there’s a row/column of all 0s det=0, if triangular then det=product of diagonal entries, if has 2 rows/columns the same det=0, if one row/column multiple of another det=0, if a is scalar det(aA)=a^ndet(A)

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17
Q

Determinants of elementary matrices

A

Swap row: -1
Multiple of row: scalar
Addition of rows: 1

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18
Q

Algebraic multiplicity

A

Number of times factor shows up in characteristic polynomial

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19
Q

If you have distinct eigenvalues w/corresponding eigenvectors…

A

They are linearly independent

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20
Q

Similar matrices

A

There exists a matrix S such that B=S^-1AS or A=SBS^-1

21
Q

Properties of similar matrices

A

Det(A)=det(B), A is invertible iff B is, A&B have same rank and characteristic polynomial, A&B have same eigenvalues

22
Q

Diagonalizable

A

Similar to diagonal matrix (that contains eigenvalues), can only happen if there are n linearly independent eigenvectors

23
Q

Matrix invertible iff

A

Geo multiplicity and algebraic are equal

24
Q

Adjoint

A

Transpose of the matrix of cofactors (find by cofactors expansion)

25
Q

If there’s an orthonormal basis you can do

A

The vector times the linear combo vectors to find the scalar multiple

26
Q

Properties of orthogonal matrices

A

QQ^T=I, Q^-1=Q^T, magnitude of Qx equals magnitude of x, det(Q)=+-1

27
Q

Properties of Orthogonal subspaces

A

If orthogonal then intersection of U and V is 0, dim(U)+dim(U^perp)=n, (U^perp)^perp is U, U intersect U^T is 0, (col(A))^perp=null(A^T), (row(A))^perp=null(A)

28
Q

Orthogonal projection of v onto subspace

A

Projection is element of U, v-p is element of U^perp

29
Q

Distance between 2 vectors

A

Magnitude of u-v

30
Q

Triangle inequality

A

Magnitude of u+v is less than/equal to magnitude of u plus magnitude of v

31
Q

Cauchy- Schwartz inequality

A

Absolute value of u dot v is less than/equal to magnitude of v times magnitude of u

32
Q

Properties of dot product/vectors

A 
v+w=w+v
u dot v= v dot u
Magnitude of u=mag of -u
udotv+udotw=udot(v+w)
udotu=mag u squared
(v+w)+u=v+(w+u)
v+0=v
u+-u=0
1u=u
a(u+v)=au+av
(a+b)u=au+bu
(ab)u=a(bu)
33
Q

Orthogonal projections (just vectors)

A

P in direction of u

v-p orthogonal to u

34
Q

How many solutions can systems of equations have?

A

0(inconsistent),1(consistent), infinite

35
Q

Matrix addition

A

Functions the same as regular addition

36
Q

Matrix multiplication

A

AB does not necessarily equal BA, AB=BC does not imply A=C, AB=0 does not mean one of the matrices is 0 matrix

37
Q

Properties of matrices

A

A(B+C)=AB+AC, (B+C)A=BA+CA, B(aA)=(aB)A, (A+B)^T=A^T+B^T, (A^T)^T=A, (aA)^T=aA^T, (AB)^T=B^TA^T

38
Q

Inverses

A

Exist if AB=BA=I for square matrices only

39
Q

Two square matrices A and B are row equivalent iff…

A

A is a product of elementary matrices times B

40
Q

Vector space

A

There is an addition and scalar multiplication that satisfies: v+w=w+v, (v+w)+u=v+(w+u), 0 vector in V such that v+0=v, for each u in V there’s -u in V such that u+-u=0, a(u+w)=au+aw, (a+b)u=au+bu, a(bu)=(ab)u, 1u=u

41
Q

Examples of vector spaces

A

F(R)- space of all real valued fctns whose domain is R, P-space of all polys in one variable, P_n-subspace of P that contains all polys of degree n, C(R)-subspace of F(R) that contains cont fctns, C[a,b]- set of real valued fctns on closed interval, R^mxn-set of mxn matrices with real entries

42
Q

Dimension of vector spaces

A

Infinite dimensional if V has no finite basis

43
Q

Linear transformation for vector space

A

v_1,v_2 in V such that T(v_1+v_2)=T(v_1)+T(v_2), T(av_1)=aT(v_1)

44
Q

Matrix of transformation for vector space

A

V goes to W with trans and [v]_E goes to [w]_F with matrix of trans

[T(v)]_F=M[v]_E

45
Q

Inner product

A

<u> is a scalar that satisfies <u> greater than/equal to 0 and <u>=0 iff u=0, <u>=, =a<u>+b</u></u></u></u></u>

46
Q

<a>=</a>

A

Tr(B^TA)

47
Q

Magnitude of matrices

A

sqrt(tr(A^TA))

48
Q

Orthogonal functions

A

=0

Linear Algebra (1 decks)

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