Pretty straightforward: is it permissible to have a garment in which linen and wool fabrics are both present? Some examples would include the example in the title (linen lining in a wool jacket) and a linen collar or lapel in a wool suit (or vice-versa).
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This issue is a dispute of Rishonim.
The Rash to Kilayim 9:1 says the wool and linen must touch to be prohibited.
The Rambam (Kilayim 10:3) holds even if they are not touching, as long as they are in the same garment it is shatnez.
Shulchan Aruch (Y.D. 299:2) rules like the Rambam.
The Rema there argues and holds like the Rash. Magen Avraham takes this position.
So if they are not touching each other, even though it is in the same garment, is a dispute.
According to the Rash and Rema, you could have your jacket if there was a separation between. According to the Rambam and Shulchan Aruch, you could not.
A linen shirt and a wool jacket would be allowed even according to the Rambam (Kilayim 10:11):
However, the Rema (300:4) brings an opinion that a garment with sleeves is different and one cannot wear a wool one over a linen one, which he says one should be careful for.
No, my mother works in the suit industry, and she asked our community rabbi this exact question. He said that as long as it is in the same garment it is impermissible; it does not need to be sewn together. I do not know the sources he used to answer her.