It's true that -we have statements about linear algebra for which (in ZFC) we can neither prove nor disprove-[1]. However, given that these statements simply aren't true, it doesn't tell us much: it's not that we can't know everything there is to know about linear algebra. It's that these things we can't know about linear algebra aren't facts about linear algebra to start with.

[1]: On a second thought, let me rephrase that: we have statements about structures partially described by linear algebra which (in ZFC) we can neither prove nor disprove for all of them at the same time.