Exam 2 (someone esle)
What is the right distributive law?
(B+C)A=BA+CA
How can you test if f(a)=f(b) holds under scalar multiplication?
(cf)a=cf(a)=cf(b)=(cf)b
How can you test if f(a)=f(b) holds under scalar addition?
(f+g)a=f(a)+g(a)=f(b)+g(b)=(f+g)b
u+(v+w)
(u+v)+w
If an entire row or column consists of zeros, what is the determinant?
0
u+-u=
0
Given v₁ and v₂ are in vector space V, let H=span{v₁,v₂}. Show H has the zero vector
0 vector is because 0v₁+0v₂=0
What is the left distributive law?
A(B+C)=AB+AC
What is the operation to create a transpose of any matrix?
Given an mxn matrix A, the transpose of A is nxm matrix whose columns are formed from the corresponding rows of A
Knowing that v₁,v₂.....vp are in a vector space V, is sufficient to know that the span{v₁.....vp} is subspace of V?
True
Is the zero vector of V a subspace?
Yes called the zero subspace {0}
Is AC=I sufficient to prove that A^-1=C?
Yes, because A=C^-1 and C=A^-1
IS every vector space a subspace?
Yes, of itself and possibly of larger spaces (like H of V)
What is a vector space?
a non empty set V of objects called vectors for which addition and multiplication of scalars subject to the 10 axioms hold.
Is a null space of an mxn matrix A a subspace of Rⁿ?
yes
What degree i p(t)=a₀?
zero
For an invertible matrix A in Rⁿ, det(αA)=
αⁿ*det(A)
Row Operations and effect on determinants: If two rows of A are interchanged to make B, then Det B=...
-Det(A)
If A and B are nxn matrices, then: det AB=
=(detA)(detB)
Prove why detA^T=detA.
A cofactor expansion along the first column of A is the same as a cofactor expansion along the first row of A^T
What is the proof for x=A^-1b?
A solution exists if A-1b is substituted for x and then Ax=A(A^-1*)b=(A*A^-1b)=Ib=b. To prove it is unique Au=b,A^-1*A*u=A^-1*b,Iu=A^-1*b,u=A^-1*b
What is a subspace?
A subset of vector space V called H where three properties hold.
What is the adjugate/adjoint matrix A?
A transpose of matrix B where each entry in matrix B is of cofactor matrix (remember to alternate positive and negatives)
What is the associative law of multiplication?
A(BC)=(AB)C
What is the equation for A^-1?
A^-1=(1/detA)*adjA
What is the det Ai(ej)?
Adjugate matrix of A
What is the nul space of A?
All solutions to the homogenous equation Ax=0
What is an elementary matrix?
An identity matrix with a single row operation
Let T:Rⁿ-->Rⁿ be a linear transformation and let A be the standard matrix for T. Then T is invertible if and only if A is___________. In that case, the linear transformation S given by S(x)=_________ is a unique equation satisfying S(T(x))=x and T(S(x))=x for all x.
An invertible matrix
Is general form to answer questions with basic variables in terms of free variables or free variables in terms of basic variables?
Basic variables in terms of free variables
Row Operations and effect on determinants: if the multiple of one row of A, is added to another row to produce matrix B, then Det(B)=...
Det(A)
If there are r interchanges of rows in getting A to echelon form (now matrix U), what is the equation for Det A?
DetA=(-1)^R detU
If A, B, and C are matrices of the same size and r and s are scalars: (A+B)+C≠A+(B+C)?
False
If A, B, and C are matrices of the same size and r and s are scalars: (r+s)A≠rA+sA?
False
Vector spaces do not behave similarly to Rⁿ?
False.
How do you find out if a point will be in a the subspace spanned by n vectors.
Find out if the point is a solution to the vectors using an augmented matrix
Suppose CA=In. Why does Ax=0 only have the trivial solution?
If a is invertible, it means that it is an nxn matrix and in order for ad-bc≠0, the no vector can be a multiple of another. Thus it is linear independent so the solution to Ax=0 can only have the trivial solution.
When is the column space of an mxn matrix A all of R^m?
If and only if there is at least ones solution to every b in Ax=b
Does AB=BA?
In general, no
What is the associative property of addition?
It doesn't matter which vector you add together first, it will always get the same answer
When you multiple A(mxn)*B(nxp), each column of AB is a ___________ using weights from the corresponding ________?
Linear Combination columns of b
Is H{[s,t,0]:s and t are real} a subspace of R³
Looks and acts like R² but is intact in R³. Three properties hold.
What is a zero matrix?
Matrix where all entries are zero
If you factor out a common denominator of a row in a span of vectors, what do you do with that factored out value?
Multiply determinant by that amount
If AB=0, does at least A or B equal 0?
No
If a plane/line in R³/R² does not go through the origin, is it a subspace in its dimension?
No because the plane/line does not contain the zero vector
Is R² a subspace of R³?
No, R³ has three entries and R² has two.
What is the difference between a singular and non-singular matrix?
Non-singular matrix-invertible Singular-non invertible
Does (det A+ det B)=Det(A+B)?
Not true in general
The column space of an mxn matrix A is a subspace of what dimension?
R^m
What is the algorithm for calculating the inverse of any matrix, A?
Row reduce [A I] to [I A^-1]
What is the column space?
Set of all linear combination of the columns of A. i.e. span{a₁,a₂.....an}
When the Nul(A) contains nonzero vectors, what two things are known about the solution?
Solution set is linearly independent because the free variables are the scalars. number of vectors in spanning set is equal to number of free variables
Given v₁ and v₂ are in vector space V, let H=span{v₁,v₂}. Show H is closed under vector addition
Take two arbitrary vectors u=s₁v₁+s₂v₂ and w=t₁v₁+t₂v₂ u+w=s₁v₁ +s₂v₂+t₁v₁+t₂v₂= (s₁+t₁)v₁+(s₂+t₂)v₂
Suppose the last column on AB is entirely 0's but B itself has no column of zeros. What can be said about the columns of A?
The columns of A are linearly dependent because the columns of B act as scalars for Matrix A. If a column of AB=0 then there exist non trivial solutions to Ax=0.
How is the degree of a polynomial set determined?
The degree of P is the highest power of t whose coefficient is not zero.
What is the inverse of an elementary matrix E?
The elementary matrix that transforms E back to I.
If AB=AC, does B=C?
This is not true in general
If A, B, and C are matrices of the same size and r and s are scalars: A+0=A?
True
If A, B, and C are matrices of the same size and r and s are scalars: A+B=B+A?
True
If A, B, and C are matrices of the same size and r and s are scalars: r(A+B)=rA+rB?
True
If A, B, and C are matrices of the same size and r and s are scalars: r(sA)=(rs)A?
True
Is the set of all solutions to the homogenous equation a subspace of Rⁿ?
YES
Can you calculate the determinant of an nxn matrix by a cofactor expansion across or down any row?
Yes
Given matrix A, B and scalar r, does r(AB)=(rA)B=A(rB)?
Yes
If AB is invertible, are both A and B necessarily invertible?
Yes
Is the product of two invertible matrices invertible?
Yes
Is det([d,-c;-b,a)=det([d,-b;-c,a])? Why or why not?
Yes because detA^T=detA
If H is a subspace of vector space V, is H a vector space?
Yes, because the if H is a subspace of V, the vector space operations already found for V apply to H.
Are elementary matrices invertible?
Yes, since all row operations are reversible and the identity matrix is invertible, elementary matrices are always invertible.
Is the product of nxn invertible matrices invertible?
Yes.
Is the detA=0 when the columns of A are linearly dependent. How about rows?
Yes. Yes. The rows of A are the columns of A^T. Linearly dependent columns make A^T singular.
When A^T is singular, is A? How about when A is nonsingular?
Yes. Yes.
If A and B are invertible, is AB?
Yes. (AB)^-1=(B^-1)(A^-1)
If A is invertible, is A^-1 invertible?
Yes. (A^-1)^-1=A
If A is an invertible matrix, is A^T?
Yes. (A^T)^-1=(A^-1)^T
If A is an invertible nxn matrix, does each b in the equation Ax=b have a unique solution?
Yes. x=A^-1b
What are the three properties that must hold for H to be a subspace of vector space V?
Zero vector of H is in V H is closed under vector addition(each u+v is in H) H is closed under vector multiplication (For each u, the vector cu is in H)
Find the inverse of [1,0,0;0,1,0,-4,0,1]
[1,0,0;0,1,0,+4,0,1]
What is the determinant of A (2x2)=?
ad-bc
What is the formula to test if a 2x2 matrix is invertible?
ad-bc≠0
For nxn matrix A and any b in Rⁿ, let Ai(b) be the matrix obtained from replacing what with what?
column i with vector b Ai(b)=[a₁,a₂....b......an]
Given v₁ and v₂ are in vector space V, let H=span{v₁,v₂}. Show H is closed under multiplication of scalars.
cu=c(s₁v₁ +s₂v₂)=(cs₁)v₁+(cs₂)v₂ This shows cu is in H and closed in H
What is det(B^-1AB)?
det(A), because det(B^-1)=1/det(B)
A vector space is a non-______set V of objects, called vectors, on which are defined two operations, _____ and ________by scalars (real numbers), subject to ten axioms.
empty addition and multiplication
What does a₁₃ refer to?
entry in first row, third column of a
Row Operations and effect on determinants: If one row of A is multiplied by constant k to create matrix B, then Det(B)=? What about det(A)
k*Det(A), 1/kdet(B)
What is a diagonal matrix?
matrix with non zero entries on the diagonal and 0 entries everywhere else
How does the nul(A) appear is set notation?
nul(A)={x:x is in Rⁿ and Ax=0}
The associative and distributive laws essentially say that pairs of _________ can be inserted and deleted without changing the end result.
parenthesis
What is Cramer's rule?
picture
If A is a triangular matrix, then det A is?
product of the entries on the main diagonal
An nxn matrix A is invertible if and only if A is ________ to In and any sequence that transforms A to In also transforms _____ into _______
row equivalent to In, In A^-1
The axioms hold for u,v, and w in V and scalars c and d:What are the 5 axioms for scalar multiplication?
scalar multiple of u by c, denoted as uc is in B c(u+v)=cu+cv (c+d)u=cu+du c(du)=(cd)u 1u=u
What is the geometric interpretation of the null of A?
set of all x in Rⁿ that are mapped to the zero vector of R^m via the linear transformation of T
If A is an mxn matrix and if B is an nxp matrix with columns b₁,b₂....bp, then the product AB is
the mxp matrix whose columns are Ab₁,Ab₂,Ab₃....
u+v=
u+v
The axioms hold for u,v, and w in V and scalars c and d: What are the 5 axioms relating to addition?
u+v is in V u+v=v+u (u+v)+w=u+(v+w) There is a zero vector in V such that u+0=u For each u in V, there is a vector -u such that u+(-u)=0
If all polynomial coefficients are zero, what is the degree?
undefined but included in Pn
