0
$\begingroup$

If $\mathtt{V}$ is a vector space over field $F$ and $W_1$ and $W_2$ are subspaces of $\mathtt{V}$, then find $W_1 \cup W_2$ if $W_1=\{ ( \alpha ,0) \,\vert\, \alpha \in F\}$ and $W_2=\{ (0,\beta) \,\vert\, \beta \in F\}$.

Check whether this union as well as intersection forms subspace of V.

Now I have difficulty in finding the union and intersection of sets as this set consists of 2-tuples.

  • 2
    Think two axes in the plane – 2017-01-02
  • 0
    That means union consists of all points lying on the lines $x= \apha$ & $y= \beta$ ? – 2017-01-02
  • 1
    no. Alpha and beta are not fixed elements. It's the axes. – 2017-01-02
  • 0
    Kindly give the final answer. – 2017-01-03
  • 1
    the union is the axes. – 2017-01-03
  • 0
    But alpha and beta are the field elements that are constants. How the union comes to be the axes? – 2017-01-03
  • 0
    Yes I got it. Need to know the intersection from you. – 2017-01-03
  • 1
    alpha and beta are not constant. $W_1$ is the first axis with second component vanishing. $W_2$ is the second axis with first component vanishing. The union of the two axes is.... two axes. – 2017-01-03
  • 1
    H_1317 has already given you the intersection – 2017-01-03
  • 0
    I got them right. Thanks – 2017-01-03

1 Answers 1

2

Try thinking about the union in the abstract more before diving into dealing with the tuple.

So for x $\in$ W1 and y $\in$ W2 what would the union look like? likewise for intersection.

The union would look like what Kavita mentioned and the intersection would be clearly (0,0) and do you see anything else?