# Linear Algebra Basis functions and matrices

I have no idea how to solve this and do not even understand the question. Excuse my lack of formatting and assume that that when a number is after a letter like $$e_2$$ the number is subscript. $$V_2$$ is a linear space with basis $$e = (e_1,e_2)$$ and the systems of vectors $$a=(a_1,a_2)$$ and $$b=(b_1,b_2)$$ where $$a_1 = e_1+ 2e_2$$, $$a_2 = e_1 + 3e_2$$, $$b_1=-e_1+e_2$$,$$b_2 = -e_1+ 2e_2$$. Let $$f$$ be a linear transformation on $$V_2$$.

1) If $$f(e)=b$$ find the matrix $$A$$ of $$f$$ in basis $$e$$ and $$f(v)$$ where $$v=3e_1-2e_2$$.

2) If $$f(b)=a$$ find the matrix $$M$$ of $$f$$ in base $$e$$

• I thnk the function f is supposed to be applied to the 2x2 matricies of e and b – waxo99 Dec 13 '19 at 19:48
• Never mind. I haven’t had my coffee yet this morning. It looks like you’re meant to read $f(e)=b$ as $f(e_1)=b_1$ and $f(e_2)=b_2$, and similarly for the second question. Use linearity to compute $f(v)$ and use the fact that the columns of a transformation matrix are the images of the basis vectors to find the matrices. – amd Dec 13 '19 at 19:48
• If I could get a detailed answer I would appreciate it but I understand if you do not have the time. – waxo99 Dec 13 '19 at 19:55

Question 1:

As amd says in his comment, we are meant to read $$f(e) = b$$ as $$f(e_j) = b_j$$ for all indices $$j$$. With that in mind, we can compute the matrix of $$f$$ with respect to $$e$$ as follows.

To find the first column of the matrix, note that $$f(e_1) = b_1 = -e_1 + e_2 = (-1)\cdot e_1 + (1) \cdot e_2.$$ It follows that $$(-1,1)$$ is the coordinate vector of $$f(e_1)$$ relative to $$e$$, and is thus the first column of our matrix. In other words, we have $$A = \pmatrix{-1&?\\1&?}.$$ Similarly, find that the second column is $$(-1,2)$$, so that the matrix is given by $$A = \pmatrix{-1&-1\\1&2}.$$ From there, we could compute $$f(v)$$ in two ways. One option is to use the linearity of $$f$$ as follows: $$f(v) = f(3e_1 - 2e_2) = 3f(e_1) - 2f(e_2) = 3b_1 - 2b_2 \\ = 3 (-e_1 + e_2) - 2(-e_1 + 2e_2) \\ = -3e_1 + 3e_2 + 2e_1 - 4e_2 \\ = -e_1 - e_2$$ The other option is to compute the transformation matrix that we just computed. Since $$v = 3e_1 - 2e_2$$, the coordinate vector of $$V$$ relative to $$e$$ is $$(3,-2)$$. Thus, the coordinate vector of $$f(v)$$ is given by $$A [v]_e = \pmatrix{-1&-1\\1&2} \pmatrix{3\\-2} = \pmatrix{-1\\-1}.$$ This is the coordinate vector of $$f(v)$$, which is to say that $$f(v) = -e_1 - e_2$$, confirming the earlier result.

Question 2:

This one is a little bit trickier. Remember that to find the matrix of $$f$$ with respect to $$e$$, we need to figure out what $$f(e_1)$$ and $$f(e_2)$$ are. We were given this information directly in the last problem, but now we have to get that information using what we know about $$f(b_1)$$ and $$f(b_2)$$.

We can use the following procedure:

• Find numbers $$x_1$$ and $$x_2$$ such that $$x_1 b_1 + x_2 b_2 = e_1$$. This is technically a system of equations, but it's easy to solve by "guessing" an answer.
• Note that $$f(e_1) = f(x_1b_1 + x_2 b_2) = x_1 f(b_1) + x_2 f(b_2)$$. The vector of coefficients is the coordinate vector of $$f(e_1)$$ relative to $$e$$, and is our first column of $$M$$.
• Do the same thing to find the second column, which is the vector of coefficients of $$f(e_2)$$.

Regarding the matrix-multiplication formula: suppose that $$P = [f]_{b \to e}$$, i.e. $$P$$ is the matrix of $$f$$ relative to the bases $$b$$ and $$e$$. For your problem, we would have $$P = \pmatrix{1&1\\2&3}.$$ Suppose that $$B = [I]_{b \to e}$$, i.e. $$B$$ is the change of basis matrix from $$b$$ to $$e$$. For your problem, we would have $$B = \pmatrix{-1&-1\\1&2}.$$ Then the matrix $$M$$ of $$f$$ relative to the basis $$e$$ would be given by $$M = [f]_{b \to e}[I]_{b \to e}^{-1} = PB^{-1} = \pmatrix{1&1\\2&3}\pmatrix{-1&-1\\1&2}^{-1} = \pmatrix{-1&0\\-1&1}.$$

• What about matrix A – waxo99 Dec 13 '19 at 20:29
• I had them mixed up, see my latest edit – Ben Grossmann Dec 13 '19 at 20:30
• What about matrix M then – waxo99 Dec 13 '19 at 20:31
• See my latest edit – Ben Grossmann Dec 13 '19 at 20:33
• Can I get a detailed answer for the second part and can I ask if in general we have f(a) and F is the matrix of the function F and A is the matrix of a system of equations does this equal F times A or A times F ? – waxo99 Dec 13 '19 at 20:51