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8) Let u, v be non-zero vectors. Prove that (u, v) is linearly dependendent if and only the vectors are scalar multiplies of one another.

8) Let u, v be non-zero vectors. Prove that (u, v) is linearly dependendent if and only the vectors are scalar multiplies of one another.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.4: Linear Transformations
Problem 24EQ
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Quesrion 8

1) Suppose v1, V2, V3, V4 span a vector space V. Prove that the list V1-V2, V2 - V3, V3 – V4, V4 also spans V. [3] [ 21 [51 2) Find a number t such that 1,-3,19 is not linearly independent in R³. 3) (a) Show that if we consider C a vector space orver R, then the list {1+ i,1- i} is linearly independent. (b) Show that if we consider Ca vector space orver C, then the list {1+i, 1- i} is linearly dependent. 4) Prove or give a counterexample: If v1, v2,... Vm is a linearly independent list of vectors in a vector space V (over either Q, R, C), then 5v, - 4v2, V2, .. Vm linearly independent. 5) Prove or give a counterexample: If v, vz,.. Vm and w1, W2, ... Wm are inearly independent lists of vectors in a vector space V (over either Q, R, C), then 5v1 + W1, V2 + W2, . Vm + Wm is linearly independent. 6) Let u, v be vectors in the space V pver the field IF and c a scalar. Prove that Span(u, v) = Span(u, cu + v). 7) Let u, v be vectors in the space V pver the field IF and c a non-zero scalar. Prove that Span(u, v) = Span(cu, v) 8) Let u, v be non-zero vectors. Prove that (u, v) is linearly dependendent if and only the vectors are scalar multiplies of one another. 9) Let V be a vector space and assume that (v1, v2, V3) is a linearly independent sequence from V, w is a vector from V, and that (v1 + w, vz + w, v3+ w) is a linearly dependent. Prove that w e Span(v1, v2, V3). is ... %3D
Transcribed Image Text:1) Suppose v1, V2, V3, V4 span a vector space V. Prove that the list V1-V2, V2 - V3, V3 – V4, V4 also spans V. [3] [ 21 [51 2) Find a number t such that 1,-3,19 is not linearly independent in R³. 3) (a) Show that if we consider C a vector space orver R, then the list {1+ i,1- i} is linearly independent. (b) Show that if we consider Ca vector space orver C, then the list {1+i, 1- i} is linearly dependent. 4) Prove or give a counterexample: If v1, v2,... Vm is a linearly independent list of vectors in a vector space V (over either Q, R, C), then 5v, - 4v2, V2, .. Vm linearly independent. 5) Prove or give a counterexample: If v, vz,.. Vm and w1, W2, ... Wm are inearly independent lists of vectors in a vector space V (over either Q, R, C), then 5v1 + W1, V2 + W2, . Vm + Wm is linearly independent. 6) Let u, v be vectors in the space V pver the field IF and c a scalar. Prove that Span(u, v) = Span(u, cu + v). 7) Let u, v be vectors in the space V pver the field IF and c a non-zero scalar. Prove that Span(u, v) = Span(cu, v) 8) Let u, v be non-zero vectors. Prove that (u, v) is linearly dependendent if and only the vectors are scalar multiplies of one another. 9) Let V be a vector space and assume that (v1, v2, V3) is a linearly independent sequence from V, w is a vector from V, and that (v1 + w, vz + w, v3+ w) is a linearly dependent. Prove that w e Span(v1, v2, V3). is ... %3D
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Since it's an if and only if proof, don't you have to prove "if linearly dependent, then scalar multiples" and then "if scalar multiple, then linearly dependent"?

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