What is meant by a transformation being linear?

Question

Im doing a calculus course and just started linear algebra and matrices. I understand most matrix rules like multiplication, reduction and so on. When it comes to linear mapping I'm completely confused. I don't understand how there transformations work especially in vector form.

If someone could help me understand maybe first what is meant by the column vector. Are these just x,y,z's and so on. And how do these transformations work, what makes it linear because they've said they can map lines to points and stuff.

And also what is meant by an image under a transformation?

Answer 1

A column vector of length $n$ is a matrix $\mathbf{v}$ which has $1$ column and $n$ rows: $$\mathbf{v} = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\x_n \end{pmatrix}$$

You can add column vectors (of the same length) together. That is, if $$\mathbf{w} = \begin{pmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{pmatrix}$$ is another column vector, then the sum $\mathbf{v} + \mathbf{w}$ is the vector $$\begin{pmatrix} x_1 + y_1 \\ x_2 + y_2 \\ \vdots \\ x_n +y_n \end{pmatrix}$$ You can also multiply a column vector by a scalar. If $\lambda$ is a scalar, then $\lambda \mathbf{v}$ is the column vector $$\begin{pmatrix} \lambda x_1 \\ \lambda x_2 \\ \vdots \\ \lambda x_n \end{pmatrix}$$ Anyway, if you consider the set $V$ of all column vectors of a fixed length $n$, and you consider the operations of column addition and scalar multiplication which I just described, then $V$ is a vector space. Now, let $m$ be another positive integer, and let $W$ be the vector space of column vectors of length $m$. Let $A$ be an $m$ by $n$ matrix whose entries are denoted $a_{ij}, 1 \leq i \leq m, 1 \leq j \leq n$. To each column vector $\mathbf{v} \in V$ of length $n$, we have an associated column vector $A\mathbf{v} \in W$ of length $m$, which is quite literally the product of the $m$ by $n$ matrix $A$ with the $n$ by $1$ matrix $\mathbf{v}$: $$A \mathbf{v} = \begin{pmatrix} a_{11} & a_{12} & \cdots &a_{1n} \\ \vdots & & \vdots\\ a_{m1} & a_{m2} &\cdots & a_{mn} \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\x_n \end{pmatrix}$$ This is the $m$ by $1$ matrix (that is, the column vector of length $m$): $$ \begin{pmatrix} a_{11}x_1 + \cdots + a_{1n}x_n \\ a_{21}x_1 + \cdots + a_{2n}x_n \\ \vdots \\ a_{m1}x_1 + \cdots + a_{mn}x_n \end{pmatrix}$$

Anyway, we have just defined a function, which we will call $T$, from $V$ to $W$, namely the function which sends each column vector $\mathbf{v}$ to the the column vector $A \mathbf{v}$. And you can check that it is linear, by which we mean $$T(\mathbf{v} + \mathbf{w}) = T(\mathbf{v}) + T(\mathbf{w})$$ (the addition on the left is addition of column vectors of length $n$, the addition on the right is addition of column vectors of length $m$) and $$T(\lambda \mathbf{v}) = \lambda T(\mathbf{v})$$ for any $\mathbf{v}, \mathbf{w} \in V$ and scalar $\lambda$. In fact, any other function $T: V \rightarrow W$ which is linear in the sense I just described, comes from some $m$ by $n$ matrix $A$.

Answer 2

A linear transformation is one where

$T(\alpha x) = \alpha T(x)$

and

$T(x + y ) = T(x) + T(y)$

That's all the is to it.

$T((x, y)) = (2x + y, y, 0)$

Is linear because $T(c(x,y)) = T((cx,cy)) = (2cx + cy, cy, 0) = c(2x + y, cy, 0) = cT((x,y))$

and $T((x,y) + (w,z)) = T((x+w, y + z)) = (2x + 2w + y + z, y + z, 0) = (2x +y,y,0) + (2w + z, z,0) = T((x,y)) + T((w,z))$

But $T((x,y)) = (2x + y, xy, 0)$ is not because

$T(c(x,y)) = (2cx + cy, c^2xy, 0) \ne (2cx + cy, cxy, 0) = cT((x,y))$.

Also $T((x,y) + (w,z)) \ne T((x,y)) + T((w,z))$

====

A trick to notice is if the transformation involves adding a scalar other than 0 it will not be linear. ($f(x) = x + c => f(2x) = 2x + c; c \ne 2c: f(x + y) = x + y + c \ne x + c + y + c$)

If the transformation involves multiplying two terms together or taking a power of a term it will not be linear. ($f(x) = x^2 => f(2x) = (2x)^2 \ne 2(x^2): f(x + y) = (x +y)^2 \ne x^2 + y^2$)

But if the transformation is only multipling terms by scalars and adding terms it is linear.

$f((x,y)) = \sum (c_ix + d_iy) \implies f(a(x,y) + e(w,z)) = \sum (c_iax + d_iay + ec_iw + d_iez) = a\sum(c_ix + d_iy) + e\sum(c_ix + d_iy) = af((x,y)) + ef((w,z))$

====

Intuitive. If you travel along a line the distance you change vertically is proportional to the distance you change horizontally. Lines are consistant. If you double the input, you double the output. If you add two inputs together you get the two outputs added together.

If you travel anything that isn't a line that isn't true any more. That's why they call them "linear".

BUT there is a caveat. In high school algebra lines could have y-intercepts which throw these proportions off. It's only the "steepness" that is linear. In linear algebra they call those type of functions "affine". They are sort of "linear" but with a constant "slide displacement to them".

Answer 3

Column vectors are elements of a vector space as represented by a column containing certain values. For instance,

$\vec{a} = \begin{pmatrix} x_{1} \\ x_{2} \\ x_{3} \end{pmatrix} \in \mathbb{R}^{3}$.

These are sometimes written transposed as a row vector, as indicated by a $T$: $\vec{a} = (x_{1}, x_{2}, x_{3})^{T}$. This is standard notation.

As for the linear maps, they are simply maps $f: V \rightarrow W$ that follow the rule that

$f(a+\lambda b) = f(a) + \lambda f(b)$ for $a, b \in V$ and (for a basic example) $\lambda \in \mathbb{R}$.

Can you see why these would be called linear?

Finally, the image under a transformation $f$ is the set of all elements mapped under $f$, or,

$f(V) = \{ f(v) : v \in V\} \subset W$.