Matrix Representation of Linear Transformation

[KT KIM]

This is where I am stuck. I studied ordered basis and coordinates vector previous to this.
of course I studied vector space, basis, linear... etc too,
However I can't understand just this part. (maybe this whole part)
Especially

this one which says [[T(b1)]]c...[[T(bn)]]c be a columns of matrix.

Can anyone please explain me how this works? I've stuck at here too long.

 

[fresh_42]

This is where I am stuck. I studied ordered basis and coordinates vector previous to this.
of course I studied vector space, basis, linear... etc too,
However I can't understand just this part. (maybe this whole part)
Especially

this one which says [[T(b1)]]c...[[T(bn)]]c be a columns of matrix.

Can anyone please explain me how this works? I've stuck at here too long.

This is how $A$ is defined: ("Define $A$ to be ...") the images of basis vectors of $V$ under the transformation $T$ expressed in coordinates of $C$ with respect to the given bases in $C$ as column vectors of $A$.

The author then shows that the so defined $A$ describes / is in accordance to / concurs / fully determines (whatever) the entire transformation $T$, as it maps any vector $v$ when expressed in the coordinates of $V$ with respect to the basis $\mathit{B}$ (RHS) onto the image $T(v)$ expressed in the coordinates of $W$ with respect to the basis $\mathit{C}$ (LHS).

EDIT: For short: The matrix $A$ of $T$ can be written as all images of basis vectors of $V$ arranged in columns.

 

[blue_leaf77]

Any vector $v\in V$ can be uniquely written as a linear combination of $\{b_1,\ldots,b_n\}$, i.e. $v = \sum_{i=1}^n \beta_i b_i$. Operating $T$ on $v$,
$$
Tv = \sum_{i=1}^n \beta_i (Tb_i)
$$
The thing inside the bracket in right side above implies that the action of $T$ on any vector in $V$ is going to be completely characterized if you know $Tb_i$ for $i=1,\ldots,n$.

Now suppose $A$ be the matrix representation of $T$. In $k^n$, $b_1 = (1,0,...,0)^T$, $b_2 = (0,1,...,0)^T$, and so on. If you multiply $A$ with $b_1= (1,0,...,0)^T$, you will get a vector in $k^m$ which equals the first column of $A$, right? Thus the first column in $A$ equals $T$ applied to $b_1$ and written in $\mathcal{C}$ basis, which is $[[T(b_1)]]_\mathcal{C}$. The similar argument goes for the other columns of $A$.

 

[zinq]

I strongly suggest you stop here and consider some simple examples with actual numbers. Nothing makes a type of symbolic calculation clearer the first time you encounter it than working through concrete examples.

Use a 2-dimensional vector space over the real numbers. Make up simple numbers that take the place of the abstract symbols in the textbook or notes you quoted for us.

Do the explicit calculation separately for each of the two things that the quote claims are equal. This will very much get you used to this kind of calculation and help you see what is going on.