Define system of linear equations. When a system of equations is consistent? Make echelon form to solve:

-2a – 3b + 4c = 5

b – 2c = 4

a + 3b – c = 2

Define linear transformation with an example.

Let A = , v = , b= , x =,

and define a transformation T : R² → R² by T(x) = Ax then

a. find T(v).

b. Find x ∈ R² whose image under T is b.

Find AB by block multiplication of the matrices.

A= B=

Find the least square solution of Ax=c where

A=, c=

and compute the associated least square error.

Determine the column of the matrix A are linearly independent where

A =

Let A = and B = . What value (s) of k, if any, will make AB=BA?

Evaluate the determinant of the matrix.

When two column vectors in R² are equal? Give an example. Compute u+3v, -u-2v where,

Prove that the two vectors u and v are perpendicular to each other if and only if the line through u is perpendicular bisector of the line segment from -u to v.

Find the eigenvalue of A =

Define null space of a matrix A. Let

then show that v belongs to the null space matrix A.

Find the equation y = a₀ + a₁ x of the least squares line that best fits the data points (2,1), (5,2), (7,3), (8,3).

Show that the solution of y_k+2 – 4y_k+1 + 3y_k = 0 are linearly independent.

Define group. Show that the set of integers is a group with respect to addition operation.