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
