Question

Find the dimension of the subspace of all vectors in set of real numbers R Superscript 4 whose first and third entries are equal.

Answer 1

Answer with explanation:

Consider a vector having 4 elements

     $\left[\begin{array}{cccc}a_{1}&a_{2}&a_{3}&a_{4}\end{array}\right]$

it's first and third elements are equal.

That is,

     $a_{1}=a_{4}$

$\left[\begin{array}{cccc}a_{1}&a_{2}&a_{3}&a_{4}\end{array}\right]\\\\R_{1}\rightarrow R_{1}-R_{3} \\\\\text{or}\\\\R_{3}\rightarrow R_{3}-R_{1}\\\\\left[\begin{array}{cccc}0&a_{2}&a_{3}&a_{4}\end{array}\right]\text{or}\left[\begin{array}{cccc}a_{1}&a_{2}&0&a_{4}\end{array}\right]$

So, Dimension of vector=3