This definition applies to linear transformations as well, and in particular for linear transformations $T\colon \mathbb$. $f$ is onto (or onto $Y$, if the codomain is not clear from context) if and only if for every $y\in Y$ there at least one $x\in X$ such that $f(x)=y$.$f$ is one-to-one if and only if for every $y\in Y$ there is at most one $x\in X$ such that $f(x)=y$ equivalently, if and only if $f(x_1)=f(x_2)$ implies $x_1=x_2$."One-to-one" and "onto" are properties of functions in general, not just linear transformations.ĭefinition. T maps $T: \mathbb R^n$ onto $\mathbb R^m $ iff the columns of A span $\mathbb R^m $. Let $T: \mathbb R^n \to \mathbb R^m $ be a linear transformation and let A be the standard matrix for T. Then there is this bit that confused be about onto: $\mathbb R^m $ is the image of at most one x in $\mathbb R^n $Īnd then, there is another theorem that states that a linear transformation is one-to-one iff the equation T(x) = 0 has only the trivial solution.
Note that this proposition says that if A A1 An then A is one to one if and only if whenever 0 n k 1ckAk it follows that each scalar ck 0. $T: \mathbb R^n \to \mathbb R^m $ is said to be one-to-one $\mathbb R^m $ if each b in In other words, v u, and T is one to one. $\mathbb R^m $ is the image of at least one x in $\mathbb R^n $ $T: \mathbb R^n \to \mathbb R^m $ is said to be onto $\mathbb R^m $ if each b in
I'll check back after class and update the question if more information is desirable. The task is determine the onto/one-to-one of to matrices) And I don't want to get a ban from uni for asking online. surjective) means that the columns of A contain a basis of R m thus you need exactly m columns of A to be linearly independent thus must be rank (A)m and of course n m. (Sorry for not posting the given matrix, but that is to specific. Would a zero-row in reduced echelon form have any effect on this? I just assumed that because it has a couple of free variables it would be onto, but that zero-row set me off a bit. The definition of onto was a little more abstract. If the vectors are lin.indep the transformation would be one-to-one. The definition of one-to-one was pretty straight forward.