Question

#5 Linear Independence and Spanning Suppose V has a basis $\beta$ = {u, v, w} (distinct vectors). Additional Assumption: char(F) $\neq$ 2. Let S = {u + v, v + w, w + u} and T = {u + v, u - v, w}. (a) Prove S is linearly independent (using the definition of independence). (b) Prove T spans V (using the definition of span). (c) Is S $\cup$ T linearly independent? Does it span V? Explain. (d) Is S $\cap$ T linearly independent? Does it span V? Explain.

          #5 Linear Independence and Spanning Suppose V has a basis $\beta$ = {u, v, w} (distinct vectors).
Additional Assumption: char(F) $\neq$ 2.
Let S = {u + v, v + w, w + u} and T = {u + v, u - v, w}.
(a) Prove S is linearly independent (using the definition of independence).
(b) Prove T spans V (using the definition of span).
(c) Is S $\cup$ T linearly independent? Does it span V? Explain.
(d) Is S $\cap$ T linearly independent? Does it span V? Explain.

#5 Linear Independence and Spanning Suppose V has a basis β = u, v, w (distinct vectors).
Additional Assumption: char(F) ≠ 2.
Let S = u + v, v + w, w + u and T = u + v, u - v, w.
(a) Prove S is linearly independent (using the definition of independence).
(b) Prove T spans V (using the definition of span).
(c) Is S ∪ T linearly independent? Does it span V? Explain.
(d) Is S ∩ T linearly independent? Does it span V? Explain.

Added by Justin H.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sam Stansfield

Step 1

Expanding the equation, we get c1u + c1v + c2v + c2w + c3w + c3u = 0. Rearranging the terms, we have (c1 + c3)u + (c1 + c2)v + (c2 + c3)w = 0. Since B = {u, v, w} is a basis, it is linearly independent. Show more…

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Text #5: Linear Independence and Spanning Suppose V has a basis B = {u, v, w} (distinct vectors). Let S = {u + v, v + w, w + u} and T = {u + v, u - v, w}. Additional Assumption: char(F) â‰ 2. (a) Prove that S is linearly independent (using the definition of independence). (b) Prove that T spans V (using the definition of span). (c) Is SUT linearly independent? Does it span V? Explain. (d) Is ST linearly independent? Does it span V? Explain.