Linear Algebra exam 2

False

each column of AB is a linear combination of the columns of B using weights from the corresponding column of A

true

every matrix transformation is a linear transformation

true

a linear transformation is a special type of function

true

a linear transformation preserves the operations of vector addition and scalar multiplication

true

A transformation T is linear if and only if T(c1v1 + c2v2) = c1T(v1) + c2T(v2) for all v1 and v2 in the domain of T and for all scalars c1 and c2.

true

A^T + B^T = (A+B)^T

False

Every linear transformation is a matrix transformation

False

If A and B are 2 x 2 with columns a1, a2, and b1 and b2, the AB = [a1b1 a2b2]

false

if A and B are 3x3 and B = [b1 b2 b3], then AB = [Ab1 + Ab2 + Ab3]

false, the domain is R^5

if A is a 3x5 matrix and T is a transformation defined by T(x) = Ax, then the domain of T is R^3

false, R^m is the codomain, the range is where we actually land

if A is an m x n matrix, then the range of the transformation x -> Ax is R^m

false

if A is an n x n matrix then the equation Ax = b has at least one solution for each b in R^n.

true

if A^T is not invertible, then A is not invertible

false, this is an existence question

if T: R^n -> R^m is a linear transformation and if c is in R^m then a uniqueness question is "is c in the range of T?"

true

if the columns of A are linearly independent, then the columns of A span R^n

true

if the columns of A span R^n, then the columns are linearly independent

true

if the equation Ax = 0 has only the trivial solution, then A is row equivalent to the n x n identity matrix

true

if the equation Ax=0 has a nontrivial solution, then A has fewer than n pivot positions

true

if the equation Ax=b has at least one solution for each b in R^n, then the solution is unique for each b

false

if the linear transformation x -> Ax maps R^n into R^n then A has n pivots

true

if there is a b in R^n such that the equation Ax=b is inconsistent, then the transformation x->Ax is not one-to-one

true

if there is an n x n matrix D such that AD = I, then there is also an n x n matrix C such that CA = I

false

the codomain of the transformations x -> Ax is the set of all linear combinations of the columns of A

true

the superposition principle is a physical description of a linear transformation

false

the transpose of a product of matrices equals the product of their transposes in the same order

true

the transpose of a sum of matrices equals the sum of their transposes

True

AB + AC = A(B+C)

true

The second row of AB is the second row of A multiplied by the right of B

false

(AB)C = (AC)B

false

(AB)^T = A^T x B^T