State cramers rule use it to solve the linear system no

Question

MIDTERM 1-

Problem 1- Consider the matrix

Find rank A, a basis for Col A and a basis for Row A

Problem 2- Compute (or if undefined say so, explaining why)

Problem 3- (a) State Cramer's Rule.

(b) Use it to solve the linear system (no credit for solving the system directly)

Problem 4- Mark each statement True or False. Justify your answers.

(a) If AB = 0 for two square matrices A, B, then either A = 0 or B = 0.

(b) The set P₂[X, Y] of all polynomials in X and Y of degree at most 2 (together with the usual addition and multiplication by a constant) is a vector space of dimension 6.

Problem 5- Let P₄ denote the vector space of polynomial of degree at most 4 (vector space together with the addition and multiplication by a constant). Consider the differentiation map D : P₄ → P₄ given by Df = f'.

(a) Show that D is linear.

Problem 6- Mark each statement True or False. Justify your answers.

(a) If there is a linear transformation T : R⁵ → V which is onto, then dim V ≥ 5.

(b) Any linearly independent set in R³ must have exactly three elements.