Determine (with proof) if given set is a subspace of the specified vector space.
The set of vectors $$ \begin{pmatrix} a+3b-c \\ 2b-4c \\ 5a + 6c\\ 0 \end{pmatrix}$$ in $\Bbb R^4$
My answer is:
The column space of an $m \times n$ matrix is a subspace of $\Bbb R^m$. $A$ is a $4 \times 3$ matrix, and the set is the column space of $A$, therefore the set is a subspace of $\Bbb R^4$.
Is that correct and sufficient enough? Thank you
Best Answer
The easy way is $$ \begin{pmatrix} a+3b-c \\ 2b-4c \\ 5a + 6c\\ 0 \end{pmatrix}= a\begin{pmatrix} 1 \\ 0\\ 5\\ 0 \end{pmatrix}+b \begin{pmatrix} 3 \\ 2 \\ 0\\ 0 \end{pmatrix}+c \begin{pmatrix} -1 \\ -4 \\ 6\\ 0 \end{pmatrix}=a{\bf{u}}+b{\bf{v}}+c{\bf{w}}$$ so the given set is spaned by $\bf{u},\bf{v}$ and $\bf{w}$ so it's a subspace.