A transformation T: Rn -> Rm is called a linear transformation if
a._________________________
b._________________________
a. T(u + v) = T(u) + T(v) for any u, v ϵRnb. T(cu) = cT(u) for any u ϵRn
If A is an m x n matrix, then
Rank(A) + _______ = n
nullity(A)
True or False
If a square matrix A is invertible, then AT is also invertible.
True
True or False
If A is a square matrix, then ATA is always a symmetric matrix.
True
True or False
If E is the elementary matrix obtained by performing an elementary row operation on In, and if A is n x n matrix, then AE is the result of performing the same row operation on A.
False
AE is the result of performing the same column operation on A
left to right, row column
True or False
If u1, u2, u3 form a basis for a subspace S, then every vector v in S can be written as a linear combination of u1, u2, u3 in exactly one way.
True
True or False
If A is a m x n matrix, then the matrix transformation TA is a linear transformation from Rm to Rn.
False
true if Rn -> Rm
Suppose that A is an n x n matrix. Write 5 different statements about A that is equivalent to the following statement.
"A is invertible"
det(A) =/= 0
rank = n
RREF of A is In
nullity = 0
A is a product of elementary matrices
If A is an n x n matrix, and k is a constant, then det(kA) = ___ x det(A).
kn
If λ is an eigenvalue of a square matrix A, then
Geometric multiplicity of λ ___ Algebraic multiplicity of λ
≤
If λ is an eigenvalue of an invertible square matrix A, then ___ must be an eigenvalue of A-1
λ-1
if A = [(A1 B), (0 A2)] is a square block triangular matrix (where A1 and A2 are both square matrices) then det(A) = ___________.
det(A1) * det(A2)
If A is a real _______ square matrix, then A must be diagonalizable. (answer cannot be identity nor zero).
symmetric
if λ1, λ2, λ3 are different eigenvalues of a square matrix A, and v1, v2, v3 are corresponding eigenvectors, then v1, v2, v3 are ______________________.
linearly independent
The n x n square matrix A is said to be diagonalizable if there is an __________ n x n matrix P such that ________________.
invertible
P-1AP= D
If A is an n x n matrix, then the sum of all eigenvalues of A (including multiplicity) is equal to ________.
trace(A)
If A is an n x n matrix, then the product of all eigenvalues of A (including multiplicity) is equal to ____________.
det(A)
If A is a n x n matrix such that A2 = In, then A must be diagonalizable and each eigenvalue of A must be either ___ or ___.
-1 or 1
True or False
If A is n x n matrix and B is obtained from A by elementary row operations, then A and B have the same eigenvalues.
False
True or False
If an n x n matrix is diagonalizable, then it is invertible.
False
the eigenvalues can be 0
True or False
If an n x n matrix has n distinct eigenvalues, then it is diagonalizable.
True
True or False
Every real symmetric n x n matrix is diagonalizable.
True
True or False
Every diagonal n x n matrix is diagonalizable.
True
True or False
If A is an n x n diagonalizable matrix, then every non-zero vector in Rn is an eigenvector of A.
False
if Ax = 2x, Ay = 3yA(x + y) = 2x + 3y
=/= λ(x + y)
True or False
If A and B are similar n x n matrices, then every eigenvector of A is also an eigenvector of B and vice versa.
False
True or False
For any square matrix A, A and AT have the same set of eigenvalues.
True
True or False
If an n x n matrix A is diagonalizable, then A has n distinct eigenvalues.
False
True or False
If all eigenvalues of a square matrix are zero, then A is the zero matrix.
False
ex. [(0 1), (0 0)]
True or False
If A and B are similar square matrices, then they have the same trace.
True
True or False
Suppose that A is a 3 x 3 matrix and its eigenvalues are λ1, λ2, λ3. If we switch the 1st row of A with the 2nd row, then the eigenvalues of the resulting matrix will be -λ1, -λ2, -λ3.
False
True or False
If the vector [(2), (1), (4)] is an eigenvector for a 3 x 3 matrix A, then [(4), (1), (16)] must be an eigenvector for the matrix A2.
