How to find the dimension of a vector space given parametric equations? Say we have the vector space $V = \{(a, b, c, d) \in\mathbb{R}^4\; |\; a + c = 0\quad \text{and}\quad b - c + 2d = 0\}$
Then how do I calculate the dimension of the space?
If possible use a similar example as this question is for an assignment. 
Please be as detailed as possible in your answer as I have done research on and off stackexchange but I am still blank as to how to approach this.
 A: The dimension of a vector space is the cardinality of the minimal generating set which is linearly independent. Now for $V$ we have $a+c=0$ and $b-c+2d=0$. Now from first condition we observe that $c$ is $dependent$ on $a$. And from the second condition we see that $b+2d=c$. Now if we assign any arbitrary value to $a$ then the value for $c$ is fixed and hence the value of $b+2d$ is fixed. Now you can assign any arbitrary value to $b$ and then the value of $d$ is fixed.
Hence $\{a,b\}$ is a $minimal$ generating set which is linearly independent.
Hence $dim\ V=2$.
A: Consider the matrix of the coefficients of the linear equations that define $V$:
$$\begin{bmatrix}1&0&1&0\\0&1&-1&2\end{bmatrix}$$
This matrix is row-reduced and has rank $2$. The rank of this matrix is the codimension of $V$, i. e. the difference between the dimension of the ambient space (here, $4$) and $\dim V$. In other words, if $V$ is a subspace of the finite dimensional vector space $E$, then:
$$\dim V+\operatorname{codim}V=\dim E. $$
Here you get $\;\dim V+2=4$, whence $\;\dim V=2$.
