không gian vecto

❚♦➳♥ ❝❛♦ ❝✃♣ ✶ ❈❤➢➡♥❣ ✸ ❑❤➠♥❣ ❣✐❛♥ ✈❡❝t➡ ❚P ❍å ❈❤Ý ▼✐♥❤✱ ♥❣➭② ✷✾ t❤➳♥❣ ✶✶ ♥➝♠ ✷✵✶✶ ❈❤➢➡♥❣ ✸ ❑❤➠♥❣ ❣✐❛♥ ✈❡❝t➡ ✸✳✶ ❑❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✈Ò ❦❤➠♥❣ ❣✐❛♥ ✈❡❝t➡ ✸✳✶✳✶ ➜Þ♥❤ ♥❣❤Ü❛ ❳Ðt t❐♣ V = ∅ ✈➭ tr➢ê♥❣ sè t❤ù❝ R. ●✐➯ sö tr➟♥ V ➤Þ♥❤ ♥❣❤Ü❛ ➤➢î❝ ❤❛✐ ♣❤Ð♣ t♦➳♥✿ P❤Ð♣ ❝é♥❣ ❤❛✐ ✈Ð❝ t➡ V × V → V, (x, y) → x + y. P❤Ð♣ ♥❤➞♥ ♠ét ✈Ð❝ t➡ ✈í✐ ♠ét sè t❤ù❝ R × V → V, (k, x) → kx✳ V ➤❣❧ ❦❤➠♥❣ ❣✐❛♥ ✈Ð❝ t➡ tr➟♥ tr➢ê♥❣ R ♥Õ✉ ✶✵ t✐Ò♥ ➤Ò s❛✉ ➤➢î❝ t❤á❛✿ ∀x, y ∈ V : x + y ∈ V. ∀x, y ∈ V : x + y = y + x. ∀x, y, z ∈ V : x + (y + z) = (x + y) + z. P❤❛♥ ❱➝♥ ❚rÞ ❚♦➳♥ ❝❛♦ ❝✃♣ ✶ ❈❤➢➡♥❣ ✸ ❑❤➠♥❣ ❣✐❛♥ ✈❡❝t➡ ✸✳✶ ❑❤➳✐ ♥✐Ö♠ ❝➡ ❜➯♥ ✈Ò ❦❤➠♥❣ ❣✐❛♥ ✈❡❝t➡ ✸✳✶✳✶ ➜Þ♥❤ ♥❣❤Ü❛ ❳Ðt t❐♣ V = ∅ ✈➭ tr➢ê♥❣ sè t❤ù❝ R. ●✐➯ sö tr➟♥ V ➤Þ♥❤ ♥❣❤Ü❛ ➤➢î❝ ❤❛✐ ♣❤Ð♣ t♦➳♥✿ P❤Ð♣ ❝é♥❣ ❤❛✐ ✈Ð❝ t➡ V × V → V, (x, y) → x + y. P❤Ð♣ ♥❤➞♥ ♠ét ✈Ð❝ t➡ ✈í✐ ♠ét sè t❤ù❝ R × V → V, (k, x) → kx✳ V ➤❣❧ ❦❤➠♥❣ ❣✐❛♥ ✈Ð❝ t➡ tr➟♥ tr➢ê♥❣ R ♥Õ✉ ✶✵ t✐Ò♥ ➤Ò s❛✉ ➤➢î❝ t❤á❛✿ ∀x, y ∈ V : x + y ∈ V. ∀x, y ∈ V : x + y = y + x. ∀x, y, z ∈ V : x + (y + z) = (x + y) + z. P❤❛♥ ❱➝♥ ❚rÞ ❚♦➳♥ ❝❛♦ ❝✃♣ ✶ ∃ 0 ∈ V : 0 + x = x + 0 = x. 0 : ♣❤➬♥ tö tr✉♥❣ ❤ß❛ ❝ñ❛ ♣❤Ð♣ +. ∀x ∈ V, ∃ − x ∈ V : x + (−x) = (−x) + x = 0. −x : ♣❤➬♥ tö ➤è✐ ①ø♥❣ ✭♣❤➬♥ tö ➤è✐✮ ❝ñ❛ x. ∀k ∈ R, x ∈ V : kx ∈ V. ∀k ∈ R, x, y ∈ V : k(x + y) = kx + ky. ∀k, l ∈ R, x ∈ V : (k + l)x = kx + lx. ∀k, l ∈ R, x ∈ V : k(lx) = (kl)x. ∀x ∈ V : 1x = x. P❤❛♥ ❱➝♥ ❚rÞ ❚♦➳♥ ❝❛♦ ❝✃♣ ✶ ∃ 0 ∈ V : 0 + x = x + 0 = x. 0 : ♣❤➬♥ tö tr✉♥❣ ❤ß❛ ❝ñ❛ ♣❤Ð♣ +. ∀x ∈ V, ∃ − x ∈ V : x + (−x) = (−x) + x = 0. −x : ♣❤➬♥ tö ➤è✐ ①ø♥❣ ✭♣❤➬♥ tö ➤è✐✮ ❝ñ❛ x. ∀k ∈ R, x ∈ V : kx ∈ V. ∀k ∈ R, x, y ∈ V : k(x + y) = kx + ky. ∀k, l ∈ R, x ∈ V : (k + l)x = kx + lx. ∀k, l ∈ R, x ∈ V : k(lx) = (kl)x. ∀x ∈ V : 1x = x. P❤❛♥ ❱➝♥ ❚rÞ ❚♦➳♥ ❝❛♦ ❝✃♣ ✶ ∃ 0 ∈ V : 0 + x = x + 0 = x. 0 : ♣❤➬♥ tö tr✉♥❣ ❤ß❛ ❝ñ❛ ♣❤Ð♣ +. ∀x ∈ V, ∃ − x ∈ V : x + (−x) = (−x) + x = 0. −x : ♣❤➬♥ tö ➤è✐ ①ø♥❣ ✭♣❤➬♥ tö ➤è✐✮ ❝ñ❛ x. ∀k ∈ R, x ∈ V : kx ∈ V. ∀k ∈ R, x, y ∈ V : k(x + y) = kx + ky. ∀k, l ∈ R, x ∈ V : (k + l)x = kx + lx. ∀k, l ∈ R, x ∈ V : k(lx) = (kl)x. ∀x ∈ V : 1x = x. P❤❛♥ ❱➝♥ ❚rÞ ❚♦➳♥ ❝❛♦ ❝✃♣ ✶ ∃ 0 ∈ V : 0 + x = x + 0 = x. 0 : ♣❤➬♥ tö tr✉♥❣ ❤ß❛ ❝ñ❛ ♣❤Ð♣ +. ∀x ∈ V, ∃ − x ∈ V : x + (−x) = (−x) + x = 0. −x : ♣❤➬♥ tö ➤è✐ ①ø♥❣ ✭♣❤➬♥ tö ➤è✐✮ ❝ñ❛ x. ∀k ∈ R, x ∈ V : kx ∈ V. ∀k ∈ R, x, y ∈ V : k(x + y) = kx + ky. ∀k, l ∈ R, x ∈ V : (k + l)x = kx + lx. ∀k, l ∈ R, x ∈ V : k(lx) = (kl)x. ∀x ∈ V : 1x = x. P❤❛♥ ❱➝♥ ❚rÞ ❚♦➳♥ ❝❛♦ ❝✃♣ ✶ ∃ 0 ∈ V : 0 + x = x + 0 = x. 0 : ♣❤➬♥ tö tr✉♥❣ ❤ß❛ ❝ñ❛ ♣❤Ð♣ +. ∀x ∈ V, ∃ − x ∈ V : x + (−x) = (−x) + x = 0. −x : ♣❤➬♥ tö ➤è✐ ①ø♥❣ ✭♣❤➬♥ tö ➤è✐✮ ❝ñ❛ x. ∀k ∈ R, x ∈ V : kx ∈ V. ∀k ∈ R, x, y ∈ V : k(x + y) = kx + ky. ∀k, l ∈ R, x ∈ V : (k + l)x = kx + lx. ∀k, l ∈ R, x ∈ V : k(lx) = (kl)x. ∀x ∈ V : 1x = x. P❤❛♥ ❱➝♥ ❚rÞ ❚♦➳♥ ❝❛♦ ❝✃♣ ✶ ∃ 0 ∈ V : 0 + x = x + 0 = x. 0 : ♣❤➬♥ tö tr✉♥❣ ❤ß❛ ❝ñ❛ ♣❤Ð♣ +. ∀x ∈ V, ∃ − x ∈ V : x + (−x) = (−x) + x = 0. −x : ♣❤➬♥ tö ➤è✐ ①ø♥❣ ✭♣❤➬♥ tö ➤è✐✮ ❝ñ❛ x. ∀k ∈ R, x ∈ V : kx ∈ V. ∀k ∈ R, x, y ∈ V : k(x + y) = kx + ky. ∀k, l ∈ R, x ∈ V : (k + l)x = kx + lx. ∀k, l ∈ R, x ∈ V : k(lx) = (kl)x. ∀x ∈ V : 1x = x. P❤❛♥ ❱➝♥ ❚rÞ ❚♦➳♥ ❝❛♦ ❝✃♣ ✶ ∃ 0 ∈ V : 0 + x = x + 0 = x. 0 : ♣❤➬♥ tö tr✉♥❣ ❤ß❛ ❝ñ❛ ♣❤Ð♣ +. ∀x ∈ V, ∃ − x ∈ V : x + (−x) = (−x) + x = 0. −x : ♣❤➬♥ tö ➤è✐ ①ø♥❣ ✭♣❤➬♥ tö ➤è✐✮ ❝ñ❛ x. ∀k ∈ R, x ∈ V : kx ∈ V. ∀k ∈ R, x, y ∈ V : k(x + y) = kx + ky. ∀k, l ∈ R, x ∈ V : (k + l)x = kx + lx. ∀k, l ∈ R, x ∈ V : k(lx) = (kl)x. ∀x ∈ V : 1x = x. P❤❛♥ ❱➝♥ ❚rÞ ❚♦➳♥ ❝❛♦ ❝✃♣ ✶