How do I solve this linear transformation problem??

Question

Let L:R^2 ------> R^2 be a linear transformation such that
L(i+2j) = -2i+3j and L(i - j) = 5i+2j
a)Determine the matrix representing the linear transformation
b) Determine the value of L(7i+5j)

please help me :(

Answer 1

a) Suppose the matrix has first row |x y|, and second row |u v|.
When this operates on the column vector |1 2| you have
|x y||1| = |x + 2y| = |-2|
|u v||2| |u + 2v| | 3|. This tells you that x + 2y = -2 and u + 2v = 3. Similarly, when the matrix operates on the column vector |1 -1| you get the two equations x - y = 5 and u - v = 2.

Now you have two sets of equations, and the solution is x = 8/3, y = -7/3, u = 7/3, and v = 1/3.

b) First, write 7i + 5j as a linear combination of i + 2j and i - j. Find a and b such that 7i + 5j = a(i + 2j) + b(i - j). Then compute L(7i + 5j) = aL(i + 2j) + bL(i - j).

Answer 2

this is been quiet a while in view that i've got carried out those, yet from what i undergo in concepts is you certainly attempt to show that this is a linear transformation with the aid of gratifying the axioms, that are: incorporates the 0 (0) vector, preserves scalar multiplicaton and preserves addition. in case you come across a contradiction with those axioms then this is not a linear transformation. uncertain if this facilitates, yet thats how i take advantage of to sparkling up linear transformation issues and has been some years in view that.