Let ?1 ⃗ , ?2 ⃗ , ?3 ⃗ be three vectors from ℝ3 such no...

Let ?1 ⃗ , ?2 ⃗ , ?3 ⃗ be three vectors from ℝ3 such no two vectors are parallel, and ?3 ⃗ is not in the plane spanned by ?1 ⃗ and ?2 ⃗ . Prove that {?1 ⃗ , ?2 ⃗ , ?3 ⃗ } forms a basis for ℝ3

Expert Solution

Colby Messinger answered 1 year ago

Let S be the set of vectors in R4 S ={(1,1,-1,2),(1,0,1,3),(1,-3,2,2),(0,-2-1,-2)} (a) State whether the vectors...

Let S be the set of vectors in R4 S ={(1,1,-1,2),(1,0,1,3),(1,-3,2,2),(0,-2-1,-2)} (a) State whether the vectors in the set are linearly independent. (b) Find the basis for S. (c) Find the dimension of the column space (rank S). (d) Find the dimension of the null space (nullity S). (e) Find the basis for the null space of S?

Given the vectors u1 = (2, −1, 3) and u2 = (1, 2, 2) find a...

Given the vectors u1 = (2, −1, 3) and u2 = (1, 2, 2) find a third vector u3 in R3 such that (a) {u1, u2, u3} spans R3 (b) {u1, u2, u3} does not span R3

Let ? = {?1, ?2, ?3, … … … … ?? } ?? ? = {?1,...

Let ? = {?1, ?2, ?3, … … … … ?? } ?? ? = {?1, ?2, ?3, … … … … ?? } Prove by induction that the number of injective functions ? from ? ?? ? is ?!

Let ?1=(1,0,1,0) ?2=(0,−1,1,−1) ?3=(1,1,1,1) be linearly independent vectors in ℝ4. a. Apply the Gram-Schmidt algorithm to...

Let ?1=(1,0,1,0) ?2=(0,−1,1,−1) ?3=(1,1,1,1) be linearly independent vectors in ℝ4. a. Apply the Gram-Schmidt algorithm to orthonormalise the vectors {?1,?2,?3} of vectors {?1,?2,?3}. b. Find a vector ?4 such that {?1,?2,?3,?4} is an orthonormal basis for ℝ4 (where ℝ4 is the Euclidean space, that is, the inner product is the dot product).

1) Determine the angle between vectors: U = <2, -3, 4> and V= <-1, 3, -2>...

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Let (R 3 , ×) be the set of 3d vectors equipped with the operation of...

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Are the vectors v1 = (1 , 2, 3), v2 = (2, 4, 6), and v3...

Are the vectors v1 = (1 , 2, 3), v2 = (2, 4, 6), and v3 = (1, 1, 3) linearly independent or dependent? Since v2 is a scalar multiple of v1, both v1 and v2 are linearly dependent, but what does that say about the linear dependence of the three vectors as a whole?

2) Let v, w, and x be vectors in Rn. a) If v is the zero...

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Let ?1, ?2, ?3 be 3 independent random variables with uniform distribution on [0, 1]. Let...

Let ?1, ?2, ?3 be 3 independent random variables with uniform distribution on [0, 1]. Let ?? be the ?-th smallest among {?1, ?2, ?3}. Find the variance of ?2, and the covariance between the median ?2 and the sample mean ? = 1 3 (?1 + ?2 + ?3).

Find a basis for R4 that contains the vectors X = (1, 2, 0, 3)⊤ and...

Find a basis for R4 that contains the vectors X = (1, 2, 0, 3)⊤ and ⊤ Y =(1,−3,5,10)T.

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