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https://texercises.com/exercise/examples-of-linear-subspaces/
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Exercise:
abcliste abc Let V be a vector space over K. Then _Vsubseteq V is a linear subspace. abc Fix bin K. Consider the subset U_b : x_x_x_in K^|x_+x_+x_b subseteq K^ bf claim: U_b subseteq K^ is a linear subspace iff b. abcliste

Solution:
abcliste abc by definition. abc Proof of Longrightarrow. Suppose U_b is a subspace of K^. If x_x_x_y_y_y_ in U_b then x_+x_+x_b y_+y_+y_b. Since U_b subseteq K^ is a subspace we have x_+y_ x_+y_ x_+y_b. But from above we also get x_+y_+x_+y_+x_+y_b+b Longrightarrow b+bb Longrightarrow b. Proof of Longleftarrow contradiction. Suppose b neq and x_x_x_y_y_y_in U_b. Then z_z_z_x_x_x_+y_y_y_ nicht in U_b da z_-z_+z_b+b neq b which implies that b must equal . abcliste
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Exercise:
abcliste abc Let V be a vector space over K. Then _Vsubseteq V is a linear subspace. abc Fix bin K. Consider the subset U_b : x_x_x_in K^|x_+x_+x_b subseteq K^ bf claim: U_b subseteq K^ is a linear subspace iff b. abcliste

Solution:
abcliste abc by definition. abc Proof of Longrightarrow. Suppose U_b is a subspace of K^. If x_x_x_y_y_y_ in U_b then x_+x_+x_b y_+y_+y_b. Since U_b subseteq K^ is a subspace we have x_+y_ x_+y_ x_+y_b. But from above we also get x_+y_+x_+y_+x_+y_b+b Longrightarrow b+bb Longrightarrow b. Proof of Longleftarrow contradiction. Suppose b neq and x_x_x_y_y_y_in U_b. Then z_z_z_x_x_x_+y_y_y_ nicht in U_b da z_-z_+z_b+b neq b which implies that b must equal . abcliste
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eth, hs22, linear subspace, lineare algebra, proof
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ENG (English)
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Proof
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