is basis for the subspace W. Step1: Start the Gram-Schmidt process from the first vector to get an orthogonal and then an orthonormal basis for W. ----- Answer: W₁ = 1 , W2 = is an orthonormal basis of W where a= b= 7₁d=[ are all integers. Step 2: Find the orthogonal projection of the vector u = on W using the Orthogonal Decomposition Theorem. -9 Answer: The orthogonal projection is i where e f= .h= Step 3: Find the distance of the vector u from W. Answer: The distance is where A= Note: A is an integer. d g= -3/2] 1 3 2 are all positive integers (and, as usual, the entries of the vector has no common divisor!)

Apply the Gram-Schmidt orthonormalization process to transform the given basis for R into an orthonormal basis. Use the vectors in the order in which they are given. B = {(4, 3), (0, 1)} u1 = U, =

-4 -{- ----6--E}} Let = , 0 find an orthogonal basis under the Euclidean inner product. = Orthogonal basis: a = Ex: 5 U2 -2 b = Ex: 5 -3 2 { x = ] -~ - B]₁ V1 0 V2 = V3 = C = Ex: 5 be a basis for R³. Use the Gram-Schmidt process to []}

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The following vectors form a basis for R’. Use these vectors in the Gram-Schmidt process to construct an orthonormal basis for R. (Enter your answers as comma-separated lists. Enter each vector in the form (x,, x2, ...).) 14 (a) (1, 1, 1), (2, 0, 1), (4, 8, 9) (b) (3, 2, 0), (1, 5, –1), (5, –1, 2)

All vectors and subspaces are in R". Check the true statements below: OA. If {v1, V2, V3} is an orthogonal basis for W, then multiplying v3 by a scalar c gives a new orthogonal basis {v1, V2, CU3}. OB. If A = QR, where Q has orthonormal columns, then R= Q" A. C. The Gram-Schmidt process produces from a linearly independent set {r1,..., Tp} an orthogonal set {v1, ..., vp} with the property that for each k, the vectors v1, . same subspace as that spanned by x1,..., Pk. ·•, Uk span the

4. (a) Let {b1, b2, b3, b4} be an orthonormal basis for a three-dimensional subspace of R6, and let u = b, - 2b2 + b3 - 2b4 and v= b, + b2 - b3 – bạ. Then = u'v . _? Show your work. (b) What is the length of u + v? Show your work.

Let W C R" be a subspace and let W- C R" be its orthogonal complement. Let S {u1,..., uk} be an orthogonal basis for W and S' The goal of these exercises is to give two different arguments for the fact that {V1,..., Vi} be an othogonal basis for W-. (*) dim W + dim W- = n. 4. Show that SUS' is an orthogonal basis for R" and deduce (*).

2- Verify that the set of vectors (-0.6, 0.8, 0), (0.8, 0.6, 0), (0, 0, 1) form an orthonormal basis for R³, then obtain an orthonormal basis from them. Express (3,7,-4) as a linear combination of the orthonormal basis. 3- Determine whether the following sets of vectors are linearly dependent or linearly independent. a) (1,1,0), (0,0,1), (0,1,1) in R³ b) (1,0,0), (2,2,0), (3,3,3) in R³

V= {[ xz +3 LX4 :-X₁ + 2x₂ + x₂ + 4xy=0 and X₁ + 3x₂ -X₂ - 2xy = 0 A Show that V is a subspace of Ru Be Find a basis for V and state dimension C. Replace your basis in part Bo with basis for V that consists of unit vectors