2. Let x = (9) 10 and H be the subspace of R³ with basis B = -{C).()} 5 Find [x]s.
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![2. Let x =
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(19)
10 and H be the subspace of R³ with basis B
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- Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}Find the orthogonal projection of 9e1 onto the subspace of ℝ4 spanned by <2, 2, 1, 0> and <-2, 2, 0, 1>.Let C2(-∞,∞)={f(x) in C(-∞,∞)|f''(x) exists for all x} be the set of differential functions. Show this is a subspace of C(-∞,∞).
- (5.4) (6) Find the projection of the vector v onto the subspace S. S = span 0 1 1 , 1 1 0 v = 3 4 2 projs v = ??10. Find a basis for the Subspace. What is the dimension of W?How do you prove that W= im T when W is a T-invariant subspace, and V=ker + W. Where V is finite dimentional, when you let T be a element V.
- 1. Is the set of all 2 x 2 matrices of the form a 1/ 1 b , where a and b may be any scalars, a vector subspace of all 2 x 2 matrices?Let W be the subspace of R3 with orthonormal basis {w1, w2}, whereGive an example of a non-zero subspace of R^4 which has dimension > 1 and does not contain any of standard basis vectors e1, e2, e3, e4.
- Find a basis for the two-dimensional subspace of R^4 defined byFind a basis for the subspace of R3 spanned by S.S = {(1, 2, 2), (−1, 0, 0), (1, 1, 1)}For each of the following parts, determine whether the statement is trueor false. Justify your claim with either a proof or a counterexample,whichever is appropriate.(a) Any finite dimensional subspace of C∞ is the solution space of ahomogeneous linear differential equation with constant coefficients. (b ) There exists a homogeneous linear differential equation with constant coefficients whose solution space has the basis { t , t2 }. (c) For any homogeneous linear differential equation with constantcoefficients, if x is a solution to the equation, so is its derivative x'.
