3A Midterm: True or False
How many rows and columns must a matrix A have in order to define a mapping from R⁴ to R⁵?
A must have 5 rows & 4 columns
Another notation for vector [-4, 3] is [-4 3]
False
For a square matrix A, vectors in Col A are orthogonal to vectors in Nul A
False
Points in the plane corresponding to [-2, 5] and [-5, 2] lie on a line through the origin
False
S is a LD set so each vector is a linear combo of the other vectors in S
False
Set contains fewer vectors than there are entries in the vectors, set is LI
False
Set in Rⁿ is LD, set contains more vectors than there are entries in each vector
False
Set span {u, v} is always visualized as a plane through the origin
False
When two linear transformations are performed one after another, the combined effect may not always be a linear transformation
False
(AB)^T=A^T*B^T
False: B^T*A^T
If A & B are square and invertible then A⁻¹B⁻¹ is the inverse of AB
False: B⁻¹(A⁻¹)
A is a 3x2 matrix so transformation cannot be one-to-one
False: Can be one-to-one, but not onto
A is diagonalizable if A=PDP⁻¹ for some matrix D and some invertible matrix P
False: D should be a diagonal matrix
Each line in Rⁿ is a one-dimensional subspace of Rⁿ
False: Dimension defined for subspace only
A is a 3x5 matrix and T is a transformation defined by T(x)=Ax, then domain of T is R³
False: Domain corresponds to number of columns (R⁵)
The determinant of a triangular matrix is the sum of the entries on the main diagonal
False: If A is a triangular matrix, then det A is the product of the entries on the main diagonal of A
If A is diagonalizable, A is invertible
False: Invertibility depends on 0 not being an eigenvalue, a diagonalizable matrix may or may not have 0 as an eigenvalue
Mapping is onto if every vector in Rⁿ maps onto some vector in R^m
False: Linear transformation is onto if codomain equals range
(AB)C=(AC)B
False: Matrix multiplication not commutative
Every linear transformation is a matrix transformation
False: Matrix transformations are linear transformations
If a set has property that u-sub-i dotted w/ u-sub-j=0 whenever i≠j then S is an orthonormal set
False: Might not be normal
Subspace of Rⁿ is any set H that (i) zero vector is in H, (ii) u, v, and u+v are in H, and (iii) c is a scalar and cu is in H
False: Much apply for EVERY u, v, & c
For any scalar c, magnitude of c*v=c*magnitude of v
False: Need absolute value of c
Not every orthogonal set in R^n is LI
False: Orthogonality implies LI
If A is an mxn matrix, range of transformation is R^m
False: R^m is only the codomain
To find eigenvalues of A, reduce A to echelon form
False: Row reduction changes eigenvalues and eigenvectors
If A and B are 3x3 and B=[b₁ b₂ b₃] then AB=[Ab₁+Ab₂+Ab₃]
False: Solution is not added
Set of all solutions of a system of m homogenous equations in n unknowns is a subspace of R^m
False: Subspace of Rⁿ
Determinant of A is the product of the pivots in any echelon form U of A, multiplied by (-1)^r where r is the number of row interchanges made during row reduction from A to U
False: This changes the determinant
det(A+B)=det A + det B
False: This is true for the product
Columns of matrix A are LI if equation Ax=0 has the trivial solution
False: Trivial solution is always the solution
If the linear transformation maps Rⁿ into Rⁿ then A has n pivots
False: We don't know anything about matrix A
If A is an nxn matrix then equation Ax=b has at least one solution for each b in Rⁿ
False: We need to know if A is invertible or something more
A is diagonalizable if and only if A has n eigenvalues, counting multiplicity
False: n eigenvectors must be linearly independent
An mxn upper triangular matrix is invertible when every position along diagonal is a pivot and nonzero. Also, it must be row equivalent to I
True
Any list of 5 real #s is a vector in R⁵
True
Columns of an invertible square matrix form a basis for Rⁿ
True
Each elementary matrix is invertible
True
Example of a linear combination of vectors v₁ and v₂ is vector 1/2(v₁)
True
Finding an eigenvector of A may be difficult, but checking it is easy
True
If A^T is not invertible then A is not invertible
True
If T: R²→R² rotates vectors about the origin through an angle, then T is a linear transformation
True
If a set B= {v₁, v₂...} is a basis for subspace H and if x=c₁v₁+c₂v₂... then c₁, c₂... are the coordinations of x relative to basis B
True
If columns of A span Rⁿ then columns are LI
True
If det A≠0 then A is invertible
True
If distance from u to v equals distance from u to -v, u and v are orthogonal
True
If equation Ax=0 has a nontrivial solution then A has fewer than n pivots
True
If equation Ax=0 has only the trivial solution then A is row equivalent to the nxn identity matrix
True
If the columns of A are LI then the columns of A span Rⁿ
True
If the equation of Ax=b has at least one solution for each b in Rⁿ then the solution is unique for each b
True
If there is a b in Rⁿ such that the equation Ax=b is inconsistent then the transformation is not one-to-one
True
If there is an nxn matrix D such that AD=I, then there is also an nxn matrix C such that CA=I
True
If x is orthogonal to every vector in subspace W, then x is in the set of all vectors orthogonal to W AKA orthogonal complement of W
True
In order for a matrix B to be inverse of A, both equations AB=I and BA=I must be true
True
Linear transformation is completely determined by its effect on the columns of the nxn identity matrix
True
Matrix A is not invertible if and only if 0 is an eigenvalue of A
True
Number c is of an eigenvalue of A if and only if equation (A-cI)x=0 has a nontrivial solution
True
Row operations don't affect linear dependence relations among columns of a matrix
True
Row replacement operation doesn't affect the determinant of a matrix
True
Second row of AB is the second row of A multiplied on the right by B
True
The orthogonal projections of y onto v is the same as the orthogonal projection of y onto cv whenever c≠0
True
Transpose of a sum of matrices equals the sum of their transposes
True
U and v are nonzero vectors, span {u, v} contains line through u and the origin
True
Vector u results when vector u-v is added to vector v
True
Dimensions of col(A) is # of pivot columns of A
True: # pivot columns = rank(A)
If A is an invertible square matrix then equation Ax=b is consistent for each b in Rⁿ
True: A⁻¹b=x
If v₁, v₂... are in Rⁿ then their span is the same as the column space of the matrix [v₁, v₂...]
True: Column space of an msn matrix is a subspace of R^m
An orthogonal matrix is invertible
True: Columns LI so invertible
The cofactor expansion of det A down a column is equal to the cofactor expansion along a row
True: Determinant of a square matrix can be computed by a cofactor expansion across any row or down any column
Dot product of u & v - dot product of v & u equals 0
True: Dot product is commutative
Columns of any 4x5 matrix are LD
True: Five columns w/ four entries means LD
If columns of A are LD then det A=0
True: If det A = 0 then the matrix is not invertible so it is not LI and must be LD
Two vectors are LD if and only if they lie on a line through the origin
True: If the zero vector is in their span then they are LD
If R^n has a basis of eigenvectors of A then A is diagonalizable
True: Need P and D
For an mxn matrix A, vectors in the null space of A are orthogonal to vectors in row space of A
True: Orthogonal complement of row space of A is null space of A
Transformation of T is linear if & only if T(c₁v₁+c₂v₂...)=c₁T(v₁)+c₂T(v₂)... for all v in the domain T and for all scalars c
True: Property of transformations
Dimensions of col(A) and nul(A) add up to # of columns of A
True: Rank Theorem
Set of p vectors spans a p-dimensional subspace H of Rⁿ, then these vectors form a basis for H
True: Rank Theorem
Linear transformation is a special type of function
True: Special properties such as T(cu+dv)=cT(u)+dT(v)
If x & y are LI & if {x, y, z} is LD, then z is in span {x,y}
True: Z can be written as linear combination of the other two so it is in their span
If x & y are LI, and if z is in their span, then {x,y,z} is LD
True: Z is a linear combination of the other two, which is LD
For any scalar c, dot product of u & (cv)=c*dot product of u & v
True: dot product is commutative
If the columns of an mxn matrix A are orthonormal then linear mapping preserves lengths
True: ||Ux||=||x||