Linear Algebra Worksheet

10 small questions. Just need “always true” or “sometimes false.” Thank you!!!!!

Linear Algebra
First and Last Midterm – Page 3 of 9
2. (30 points) Determine whether or not the following statements are always true or
sometimes false. Recall P if and only if Q means that if statement P is true, then
statement Q is true, and if statement Q is true, then statement P is true.
(a) (3 points) Anxn matrix A is invertible if and only if dim(Null(A))
=n.
(b) (3 points) A square matrix A is invertible if and only if det(A) > 0.
(c) (3 points) If we multiply an m x n matrix by an n xp matrix, the result is a p x m
matrix.
(d) (3 points) The rank of a matrix A is equal to the number of pivot columns it has.
(e) (3 points) Given an m x n matrix A, m = rank(A) + dim Null(A).
(f) (3 points) The columns of an nxn matrix A span R” if and only if they are linearly
independent.
(g) (3 points) A linear transformation is injective if and only if Ker(f) = {0}.
(h) (3 points) An n x n matrix A is invertible if and only if it is row equivalent to the
n x n identity matrix In.
(i) (3 points) Given an m x
matrix A, Col(A) is a subspace of R”.
(j) (3 points) A linear transformation must send 7 to 0.
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