5. (15 pts) Let 2 (a) Verify that W is a linear subspace of R4 1....
(5 points) Let 5 -4 v= 1-3 -3 and let W the subspace of R4 spanned by ū and 7. Find a basis of W?, the orthogonal complement of W in R4.
Problem 4 Let W a subspace of R4 with a set of basis: 1 [01 [2] 0 11 lo lo] Li Find and orthonormal basis for W!
Problem 4 Let W a subspace of R4 with a set of basis: 1 [01 [2] 0 11 lo lo] Li Find and orthonormal basis for W!
(1 point) Let u = 1. VE L . and let W the subspace of R4 spanned by {u, v}. Find a basis for WI. Answer:
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Let v = , u = , and let W the subspace of R4 spanned by v and u. Find a basis of W .
Find a basis for and the dimension of the subspace w of R4. W = {(3s - t, s, t, s): s and t are real numbers) (a) a basis for the subspace w of R4 (b) the dimension of the subspace W of R4
(a) Find an orthonormal basis for the linear subspace V of R4 generated by the vectors 1 1 1 1 2 (b) What is the projection of the vector on the linear subspace V?
Let W be the subspace of R4 spanned by the orthogonal vectors 1 0 0 ui , ua : 0 1 Find the orthogonal decomposition of v = ܝܬ ܥ 5 -4 6 with respect to W. -5 p= projw (v) = q= perpw («) =
Let and let T : R4 → R4 be the map T(v) = v-2 proju,V-2 proju, w (a) Show that T is a linear transformation. (b) Find T] and (c) Show that T is invertible and find T-1
Let and let T : R4 → R4 be the map T(v) = v-2 proju,V-2 proju, w (a) Show that T is a linear transformation. (b) Find T] and (c) Show that T is invertible and find T-1
5. Exercise A5: Given {ui,..., up an orthogonal basis for a subspace W of R". Let T: RnR be defined by T(x)prox, the projection of x onto the subspace W (a) Verify that T is a linear transformation. (b) What is ker(T), the kernel of T? c) What is T (R"), the range of T?
5. The given vectors form a basis for a subspace W of R3 or R4. Apply the Gram- Schmidt Process to obtain an orthogonal basis for W 2 3 1 W1 = W2 W3