# Finding a subspace which includes the basis of a solution space.

Let $$V$$ be the solution space of the following homogeneous linear system: \begin{align} x_1 − x_2 − 2x_3 + 2x_4 − 3x_5 &= 0\\ x_1 − x_2 − x_3 + x_4 − 2x_5 &= 0.\end{align} Find $$\dim(V)$$ and a subspace $$W$$ of $$\mathbb R^5$$ such that $$W$$ contains $$V$$ and $$\dim(W) = 4$$. Justify your answer.

Not sure how to go about doing this.

• "$V$ is a basis of $S$." There are many bases. Mar 16, 2020 at 6:08
• The zero vector makes any set of vectors linearly dependent. Also, it's an element of every subspace, so you wouldn’t be adding anything.
– amd
Mar 16, 2020 at 6:38
• Have updated the question accordingly, thank you for the input Mar 16, 2020 at 6:49

Let $$A=\begin{bmatrix}1&-1&-2&2&-3\\1&-1&-1&1&-2\end{bmatrix}$$ The rows of $$A$$ are linearly independent, so the rank of $$A$$ is 2. Since A has 5 columns, the dimension of $$V$$, which is the solution space of $$A$$, is 5-2=3. Let $$B=[1,-1,-2,2,-3].$$ Let $$W$$ be the solution space of $$B.$$ $$B$$ has rank 1, so the dimesion of $$W$$ is 5-1=4. Any solution of both of the given equaions is a soluion of the first equation, so $$V$$ is a subspace of $$W.$$