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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