) Let z be a particular solution to a linear equation system Az = b. Let {u, u2, uz} %3D be a spanning set for the null space N(A). Let U := (u1 u2 u3. Show that y is a solution to Az = b if and only if y = z+Uc for some e e R. SO Determine whether the following vectors in P3 are linearly independent. Justify your answers. 1² + 2r, z+1, 3

Q: 1. If A = find A1000 by diagonalizing. )Show every single step for your work).

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Q: V = {ax³ + ax² + bx − 2c|a + c = 0, b ≤ R} ≤ P3.

A: Solution

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A: Calculation:Since it is given that v and w are orthogonal, v ∙ w = 0.

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Q: 10. The set of vectors: [0 1-3 -1], (1 01 1] is linearly independent. Answer: yes or no

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Q: Q4. Let vi,..., vE E R". Assume that Span{vi,..., T} = R". In this problem you will show that n < k,…

A: Solution (a): Given that  v1→=v11v12⋮v1n, v2→=v21v22⋮v2n, …, vk→=vk1vk2⋮vkn                  . . .…

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A: By using rref of matrix we solve the given problem as follows :

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Q: The vectors V= 5 El 6 -17 u= are linearly independent if and only if k 4 8 -5+k , and w = 2 8 -18

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Q: V = {ax² + bx + 3|a, b ≤ R} ≤ P2.

A: Given: V=ax2+bx+3 | a, b∈ℝ⊂ℙ2 To write: The given set as span of minimum number of vectors.

Q: Show that the set S= {(20, 34), (40, 68)} is linearly independent or not?

A: Given set is S= {(20, 34), (40, 68)}

Q: 2. Let V₁ a. 14 -2' b. V₂ = " 16 10 spanned by B. Note that B is an orthogonal set. -49- 35 7 -28_…

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Q: Q1. Which equation can show the "linearity" of Ordinary Least Square (OLS) assumption? A) Y=B1 +…

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A: Step 1: Reference: Vector Calculus, Vector triple product

Q: 2. Consider the collection of vectors --0·00) 2 S = Is S linearly independent or linearly dependent?…

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Q: b. where 17. Consider a matrix A of the form A -a a? + b? v and w such that Av entries of v and w in…

A: A=abb-a,   a2+b2=1Let V→=xyA V→=V→⇒abb-axy=xy⇒a-1bb-a-1xy=00This is a system homogeneous linear…