help : vector spaces

[eager verge]

i got stuck in this exercise
i solved questions 1,2,3
⚪i think my solution of question 1 is correct

⚪however for question 2 :
in the part of checking if B1 is free idk if i did it correctly since i didnt change anything in the matrix i directly concluded that rang (B1)=2 = card (b1)

⚪ for question 3 : i'm so stuck
in the second condition i'm supposed to find that dim( E∩F) = 0 but i found it 1 soo i'm probably wrong
and the first condition i found that rang b = 3 and its not 4 (card b=4)
so i couldnt prove that it's a free family

i couldnt solve the rest of the exercise plsss anyone help me

[viscid jetty]

I have a feeling this question is wrong. In all the ways I've seen direct sums defined, one requirement is that E+F should reach any vector in R^3 as a unique combination of vectors of E and F. It's not the case here.

In particular, a 3-dimensional vector space cannot be the direct sum of two 2-dimensional vector spaces.

Maybe it'll depend on how it's defined in your course, but if it's defined differently then it's a poor choice of notation.

[eager verge]

i guess its defined the same way yeah

we say E is a direct sum (F⊕G) it means F∩G={0} and F+G=E

right?