LINEAR ALGEBRA FINAL EXAM QUESTIONS AND ANSWERS

LINEAR ALGEBRA FINAL EXAM
QUESTIONS AND ANSWERS
We have a system of linear equations with 15 variables and 15 equations. What are the
possible number of solutions? - Answer-0 or 1 or infinity

Let A ∈ R^(5 x 4). If Ax = 0 has the unique solution x = 0, then Ax = b has at most one
solution for any b ∈ R^5. - Answer-true

Let A ∈ R^(n x m) and x1, ..., xk ∈ R^m. If Ax1, ..., Axk ∈ R^n are linearly independent,
then x1, ..., xk are also linearly independent. - Answer-true

The column vectors of an 8 x 3 matrix are necessarily linearly independent. - Answer-
false

If the n x n matrices A and B are invertible then 2A + B is also invertible. - Answer-false

Let A ∈ R^(m x n). The set {x ∈ R^n: Ax = 0} is a subspace of R^n. - Answer-true

A is a square matrix. Then det(A) < 0, implies that A is invertible. - Answer-true

If the square matrix A is not invertible then the only eigenvalue of A is 0. - Answer-false

For invertible n x n matrices A,B we must have det(ABA^-1) = det(B). - Answer-true

Let A be an n x n matrix with integer entries, and determinant -1. Let b be an n-vector
with integer entries. The solution of Ax = b is necessarily a vector x with integer entries.
- Answer-true

If an n x n matrix has n - 1 different real eigen values (and no complex eigen values)
then it is not diagonalizable. - Answer-false

If the square matrix A is not invertible then 0 is an eigen value of A. - Answer-true

if m = n, - Answer-there exits 1 solution

Gauss-Jordan elimination - Answer-entries directly above pivots are all 0

definition of linear independence - Answer-u1, ..., um ∈ R^n linearly independent if c1u1
+ ... + cmum = 0 ---> c1 = c2 = ... = cm = 0

if linearly independent and n = m, - Answer-linearly independence <---> span R^n

span definition - Answer-span(u1, ..., um) = {all linear combinations of u1, .., um}