Solving Linear Transformations in R2: Step by Step Guide

[Fernando Revilla]

I quote a question from Yahoo! Answers

Could anyone give me a step by step method on how to solve these type of questions please? Very confused and need to know for my exams.

Let V denote the real vector space R2 and phi: V -> V be a real linear transformation such that phi((1,0)) = (4,5) and phi((0,1)) = (9,11). Express the imagine phi((x,y)) of (x,y) in terms of x and y.

Assume that w1 = (3,5) and w2 = (10,17) form an ordered basis B for V. Working from the definition determine the matrix M(subscript and superscript B) (phi) of phi with respect to the basis B.

I have given a link to the topic there so the OP can see my response.

 

[Fernando Revilla]

Using that $\phi$ is a linear map:
$$\phi(x,y)=\phi[x(1,0)+y(0,1)]=x\phi (1,0)+y\phi(0,1)\\=x(4,5)+y(9,11)=(4x+9y,5x+11y)$$
On the other hand:
$$\phi(w_1)=\phi (3,5)=(4\cdot 3+9\cdot 5,\;5\cdot 3+11\cdot 5)=(57,70)$$
Now, write $\phi(w_1)=(57,70)=\alpha_1w_1+\alpha_2 w_2$ and you'll easily find $\alpha_1$ and $\alpha_2$ by means of a simple system. In a similar way, you'll get $\phi(w_2)=\beta_1w_1+\beta_2 w_2$. Then,
$$M_B^B= \begin{bmatrix}{\alpha_1}&{\beta_1}\\{\alpha_2}&{ \beta_2}\end{bmatrix}=\ldots$$