T/F LA final
det(kA)=?
=k^n det(A)
For square matrices A and B of the same size (A - B)^2 = A^2 - 2AB + B^2
False (A - B)^2 = A^2 - AB - BA + B^2
Eigenvectors can be 0
False Eigen vectors are alwaus non-zero
the vectors [1,2,1][1,0,0][0,1,2] are linearly independent over Z^3
False by casting out algorithm the matrix has echelon form with the third vector as redundant. If the answer had not been over Z^3 it would have been True.
Every set of 6 vectors in R^7 spans R7
False every spanning set has at least 7 vectors
Every set of vectors that spans R7 is linearly independent.
False for example a basis plus one more vector spans r7 but is not linearly independent.
For all non-zero vectors u,v,w in R^n it is true that (u dot v) dot w = u dot (v dot w).
False for example u=[1,0]T v =[1,1]T and w = [0,1]T then (u dot v) dot w = 1 dot w = [0,1]T and u dot (v dot w) = u dot 1 = [1,0]T
If v and w are eigenvectors for the matrix A then v+w is also an eigenvector for a.
False for example v=[1,0]T and w=[0,1]T for [[1,0]T,[0,2]T] but v+w =[1,1]T is not.
A consisten system of 6 linear equations in 6 variables over the real numbers always has exactly one solution
False if the rank is < than 6 the system will have one or more parameters and therefore infinitely many sols
If a system of 5 linear equations in 4 variables is consistent and has rank 3 then the general solution has two parameters
False it has 1 parameter
Let B be an invertible n×n-matrix and k != a scalar. Then(kB)−1=k(B−1)
False it has 4 or less elements
A scalar λ is an eigen value for the n x n matrix A if and only if det(A-λI)!=0
False it is an eigenvalue iif det(A-λI)=0
The purpose of Heron's formula is to calculate the area of a circle.
False it is to calculate the area of a triangle.
Determinant
The scaling factor that a linear transformation changes any area in space.
Equation to find Det(2x2 matrix)
ad-bc
Negative Determinants:
"Invert" space. If space is a piece of paper, it's like flipping the page over. Also called inverting orientation.
Det(Identity matrix)=?
1
If A is a square matrix then A^T = A
False
If A,B,C are matrices of the same size then their product ABC is invertible.
False
The equation 2x+y^6/3=1 is linear
False
Given a Hill cipher with encryption matrix A, the decryption matrix is also A
False its A^-1
If v1 and v2 are linearly independent eigenvectors of A, then they correspond to distinct eigenvalues.
False see ex 8.23 where an eigenvalue has two linearly independent eigenvectors
There exists integers x and y such that 28x + 12y = 1
False since gcd(28,12) = 2, there exist integers x,y such that 28x+12y=2. However since the left hand side is even and right is odd there do not exist integers x,y such that 28+12y =1
The angle between vectors w = [-2, 1, 4, -4] and u = [3, 4, 2, 1] is greater than 90 degrees.
False the angle is acute as the dot product of w and u = 3 which is positive.
Given a system of equations A if the associated homogeneous system B has an infinite number of solutions, then A will have an infinite number of solutions
False the system A may be inconsistent.
Given a 3x4 augmented matrix, R2<-3R1+R3
False there are only 3 types of row operations and none involve 3 rows.
If u and v are vectors in R3 then u x v = v x u
False u x v = -v x u
if u and v are unit vectors in R3 then u x v is a unit vector.
False ||u x v|| is the area of the parallelogram spanned by u and v. Unless u and v are orthogonal, this area is less than 1, so u x v is not a unit vector. For example, for u = [1,0,0]T and v = [1,0,0]T we have u x v = 0.
Every linearly independent set of vectors in R7 has 7 or more elements
False, for example the first 6 vectors in a basis of R^7 are linearly independent.
A homogeneous system of equations with more equations than variables will always have at least one parameter in its solution.
False, it can happen that all variables are pivot variables in which case there are no parameters.
The matrix A = [0,1]T,[1,0]T describes a reflection about the x-axis.
False, it describes a reflection about the line x=y.
The rank of a matrix is equal to the number of its non zero rows
False, only true to matrixes in echelon form.
if u1,u2.u3 are linearly dependent vectors in R4 then they can be extended to a basis of u1,u2,u3,v of R^4
False, since the 3 init vectors are linearly dependent the set of four vectors is also linearly dependent.
The eigen values of a matrix A are its main diagonal entries
False, they are the roots of the characteristic polynomial except in special cases (triangular matrix) this will not be the same thing as the diagonal entries of the matrix.
Two planes in 3 dimensional space can intersect at a point
False, they can intersect on a lone or a point
Let A,B be 3x3 matrices. If AB=0 then A=0 pr B=0
False.
The set of vectors in R^3 where exactly two of the components are zero is a subspace of R^3
False. For example v=[1,0,0]T and u=[0,1,0]T both belong to this set, but v+u =[1,1,0]T does not. Therefore, the set is not closed under addition, so not a subspace.
if u1 , u2, u3 and v are four different vectors in R3 then v can be written as a linear combination of u1,u2,u3.
False. For example, v = [0,0,1]T ,cannot be written as a linear combo of u1=[1,0,0]T,u2=[0,1,0]T,u3 =[1,1,0]T.
A homogeneous system if 5 equations with 5 variables only has the trivial solution.
False. If there are free variable, the system has non-trival solutions.
If A is a square matrix then its column space and row space have the same basis.
False. They have the same dimension but they might not be the same.
Let u be a fixed, non-zero vector. Then the function T(v) = v + u is a linear transformation.
False. We have T(0) = u != 0, so T does not preserve the origin.
The Determinant 0 means:
That the linear transformation squished all space into a line/point, making all areas 0.
Det(M1M2) = det(M1)Det(M2)
True
For vectors u,v in R^3 the area of the parallelogram spanned by u and v is ||u x v||
True
Four invertivle square matrices A,B,C,D of the same size, (ABCD)^-1=(CD)^-1(AB)^-1
True
If A^-1 and B^-1 both exist and A^-1B^-1=B^-1A^-1 then AB=BA
True
Let A,B,C be 2x2 matrices, if A is invertible and AB=AC then B=C
True
The inverse of an elementary matrix is an elementary matrix
True
The following system of linear equations over the real numbers is inconsistent: 1x+ 2y+ 0z= 1 2x+ 3y+−1z= 1 0x+ 1y+ 1z= 2
True The echelon form is 1x+ 2y+ 0z = 1 0x+ 1y+ 1z = 2 0x+ 0y+ 0z = 1
If A is a diagonal matrix then the first standard basis vector e1 is an eigenvector of A.
True Ae1 is the first colimn of A which is a scalar multiple of e since A is diagonal.
A system of linear equations with 3 equations and 4 variables can have 4 parameters in its general solution.
True This happens when all coefficients and constant terms are 0.
If A is a matrix of size 5 x 4 and has a null space of dimension 1, then the matrix has rank 3
True by rank-nullity theorem. Rank(A)+Nullity(A)=n, and 1+3 = n which is 4
The intersection of 3 different planes in a 3d space can be a line.
True for example the planes x=0 and y=0 and x+y= 0 intersect in a line.
If A is diagonalizable n x n matrix then A has n linearly independent eigenvectors
True see theorem 8.21 which says An n×n -matrix A is diagonalizable if and only if A has n linearly independent eigenvectors. Moreover, in this case, let P be the invertible matrix whose columns are n linearly independent eigenvectors of A , and let D be the diagonal matrix whose diagonal entries are the corresponding eigenvalues. Then P−1AP = D
The set Z(23) of integers modulo 23 is a field
True since 23 is prime
Every Linear transformation T satisfies T(0) = 0
True since T(0)=T(0dot0)=0T(0)=0.
Recall that an enchanted square is a square with the sum of all rows cols and diagonals equal to the same value. The 2 x 2 enchanted squares have a basis of dimension 1.
True the 2 x 2 enchanted square is a a a a
I A is an n x m matrix of rank n and b is a vector of dimension n then the equation Av = b has at least 1 sol
True the reduced echelon form of A havs n pivot entries so there is no row of 0s at the bottom, so there isn't a 0 entry.
If v is an eigenvector for A and k!=0 is a scalar then the vector kv is also an eigenvecter for A
True we have kv!= and A(kv)=k(Av)=k(λv)=λ(kv). Therefore kv satisfies the definition of a eigenvector.
Z59 is a field
True, 59 is prime.
If one system of queations can be transformed into another system of equations using elementary row operations, then the two systems have the same set of solutions.
True, Elementary row operations do not change the set of solutions.
A matrix A is invertible if the determinant is not 0.
True, Since A is not invertible if and only if the determinant is 0.
Consider a system of 7 linear equations in 4 variables over the real numbers. Assume the system has infinite many solutions then the systems reduced echelon form cntains at least 4 rows of zeros.
True, There is at least 1 free variable, so at most 4 pivot variables. Therefore the reduced echelon form contains at most 3 non-0 rows, so at least 4 rows of zeros.
There exists a set of 7 vectors that span R7
True, a basis
Every linear transformation is a matrix transformation.
True, by Theorem 6.5.
R^4 is a subspace of R^4
True, it contains the 0 vector and it is closed under addition and scalar mult. Therefore it is a subspace.
The points (-2,0), (-1,-1),(0,-1) and (2,2) can all satisfy the same quadratic polynomial of the form y=ax^2+bx+c
True, substituting each of the four points (x,y) into the equation ax^2+bx+c=y we get the system of linear equations. When we solve it we get a=1/2 b=1/2 and c=-1
If a square matrix has a row that is a multiple of another row, then it is not inverible
True, the inversion algorithm will produce a row of 0s
Every set of vectors that spans R^5 has 5 or more vectors.
True, the set of vectors in R^3 where exactly two of the components are zero is a subspace of R^3
The vectors u= [3,2,5]T and v=[2,2,-2]T are orthogonal
True, we have v dot v = 6+4-10=0 so orthogonal. Note, positive dot is an acute angle while negative is obtuse.
Given the v basis B= {[1,0,0]T,[2,1,0]T, [1,1,1]T} of R^3 the vector with coordinates [v]B = [0,1,2] is v=[4,3,2].
True, we have v=0[1,0,0]+1[2,1,0]+2[1,1,1]=[4,3,2]
Eigenvalues can be zero
True.
Every linearly independent set of 7 vectors in R7 spans R7.
True.
If A is a diagonal matrix then the determinate of a is equal to the product of its diagonal entries.
True.
If A is an NxN matrix then the homogeneous system Av=0 has non-trivial solutions iif det(A)=0
True.
If a square matrix has three equal columns then its determinant is 0.
True.
Let A be any nxn matrix, Then A is diagonalizable iif A has n linearly independent eigenvectors.
True.
The eigenvalues of a diagonal matrix are its diagonal entries.
True. Prop 8.16 That days if A is an upper or lower diagonal matrix the eigenvalues are the entries on the main diagonal.
If A has a row that is a scalar mult of another row or a col that is a scalar mult of another colum then
det(A)=0
det(AB) = ?
det(A)det(B)
Swapping two rows of a matrix makes det(M)=
det(M)*-1
Multiplying a row of a matrix by a scalar k makes det(M)=
det(M)*k
det(Atransposed) =?
det(a)
If a has a row/col of 0s, det(A)=?
det(a) = 0
A is invertible iif ...
det(a)!=0 If this is the case then det(A^-1)= 1/det(A)
If A and B are invertible matrices the (AB)^-1 = A^-1 * B^-1
false (AB)^-1 = B^-1 * A^-1
For all the vectors u,v in R^n we have ||u||+||v||=||u+v||
false for example u=[1,0]T and v=[-1,0]T then ||u||=1 ||v||=1 and ||u+v||=0
For vectors u, v in R^3 the vectors u + v and u - v are always orthogonal
false for example when v = 0 both vectors are = u.
if u and v are vectors in R^3 then u · v = -v · u
false we have u · v = v · u
