Prove Transformation is one-to-one iff it carries linearly independent subsets of $V$ onto Lin. Ind. subsets of $W$.
112231
Showing that a one-to-one linear transformation maps a linearly independent set onto a linearly independent set
94405
Let $W$ be the set of all linear combinations of $v_1,...,v_n$. Then $W$ is a subspace of $V$.
75059
Let $T:\mathbb{R}^n\to \mathbb{R}^n$ be a linear transformation...
102168
Prove that there exists disjoint open subsets $U,V$ of $X$ such that $x \in U$ and $B \subseteq V$.
96354
Linear independence of $\sin(x)$, $\sin(2x)$, $\sin(3x)$ in Map($\mathbb{R},\mathbb{R}$)
48765
Let $S$ and $T$ be finite non-empty sets such that $|S| = |T|$. Show that the function $f : S\to T$ is onto if and only if it is one-to-one.
132487
Proving that if $V$ is an inner product space and $U_1, U_2$ are subsets, then $U_1 \subset U_2$ if and only if $U^{\perp}_2 \subset U^{\perp}_1$
10181
Entire one-to-one functions are linear
109228
Prove that $v, Tv, T^2v, ... , T^{m-1}v$ is linearly independent
140799
Prove that there exist uncountable family of subsets of $\mathbb{N}$ which is linearly (totally) ordered by relation of inclusion.
94077
Proof of linear independence of $e^{at}$
97049
If $v_1,...,v_m$ are linearly independent, then the span $v_1+w,...,v_m+w$ has dimension $\ge m-1$
64395
If $v_1,...,v_r$ are eigenvectors that correspond to distinct eigenvalues, then they are linearly independent.
121993
Prove that $f$ is one-to-one iff $f \circ h = f \circ k$ implies $h = k$.
27478
Proof that two basis of a vector space have the same cardinality in the infinite-dimensional case
62065
Prove that $v_1, \dots v_n$ is a basis of V.
55752
Prove that $n+1$ vectors in $\mathbb{R}^n$ cannot be linearly independent