enter image description hereGood morning

If we have $X_1,X_2,\cdots,X_m$ ,$m$ vectors spaces,and each vector $v_j\in Xj$, and the notation $\langle \rangle$ mean span.

Then what mean this notation please :

$$X_k \not\subset \langle \{v_j:j\neq k\}\rangle.$$ Does it mean that vector space $X_k$ not belong to the span generate by each vector $v_j$ such that $j\neq k$, or not belong to the span generate by all vectors $v_j$. Examle $m=4$ so $$X_1 \not\subset \langle \{v_2,v_3,v_4\}\rangle \text{ and } X_2 \not\subset \langle \{v_1,v_3,v_4\}\rangle \text{ and } X_3 \not\subset \langle \{v_1,v_2,v_4\}\rangle\text{ and } X_4 \not\subset \langle \{v_1,v_2,v_3\}\rangle.$$ or $$X_1 \not\subset \langle \{v_2\}\rangle \text{ and } X_2 \not\subset \langle \{v_1\}\rangle.$$ and $$X_1 \not\subset \langle \{v_3\}\rangle.$$ and $$X_1 \not\subset \langle \{v_4\}\rangle.$$

Which of them is correct, please.

  • $\begingroup$ How could the span be all vectors, if they gave you the predicate $j \neq k$? Also, your examples (before and after "or") seem to be the same. $\endgroup$ – Zhengqun Koo Jul 23 '17 at 11:19
  • $\begingroup$ It was a mistake and i have corrected. $\endgroup$ – Mokh Tar Bou Jul 23 '17 at 11:20
  • $\begingroup$ Excuse me now the question is right $\endgroup$ – Mokh Tar Bou Jul 23 '17 at 11:22
  • $\begingroup$ The examples still seem the same to me. $((A \not\subset B)\ \wedge\ (A \not\subset C)) \longrightarrow (A \not\subset (B \cup C))$. $\endgroup$ – Zhengqun Koo Jul 23 '17 at 11:24
  • $\begingroup$ my question is $X_1$ does not belong to the span generated by all vectors $v_1$,$v_2$,$v_3$ or the idivudial span $\endgroup$ – Mokh Tar Bou Jul 23 '17 at 11:26

The set $\{ v_{j} : j \neq k\}$ is the following set $$S=\{v_{1},v_{2},\dots,v_{k-1},v_{k+1},\dots,v_{m-1},v_{m}\}$$

So you have that the vector space $X_{k} \not\subset \langle S \rangle$.

Furthermore, you will have that $$X_{k} \not\subset \langle v_{i} \rangle \ \forall i \neq k,$$ because if it was a subset of these "spans", then it will be contained in the span of S. But see that this is derived from the definition of $S$ and the definition is that $$S=\{v_{1},v_{2},\dots,v_{k-1},v_{k+1},\dots,v_{m-1},v_{m}\}.$$

But you could deduce from $$X_{k} \not\subset \langle v_{i} \rangle \ \forall i \neq k$$ that $X_{k} \not\subset \langle S \rangle$.

I think you are asking about the definiton of the set $S$.

  • $\begingroup$ yes about the definition of the set $\{v_i:i\neq j\}$ mean all vector except $v_j$ $\endgroup$ – Mokh Tar Bou Jul 23 '17 at 11:34
  • $\begingroup$ and thank you so much i will add pictur to see from were i have this problem $\endgroup$ – Mokh Tar Bou Jul 23 '17 at 11:36
  • $\begingroup$ @MokhTarBou Yes, it is the first option of that you propose. $\endgroup$ – DrinkingDonuts Jul 23 '17 at 11:46
  • $\begingroup$ ok thank you so much for your help. $\endgroup$ – Mokh Tar Bou Jul 23 '17 at 11:49

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