College

Let V = R", n > 2 and let W = span B, where B = {ui, u2} and let B be orthonormal. Let A E M1,2(R) such that A = (ui|u2). State the formula for the projection of a vector x EV onto the subspace W. 5) Let V = R" and let W = col A, where A e Mnm (n > m). Assume that ATA Im (such matrices are called partial isometries). State the formula for the projection of a vector xe V onto the subspace W. 6) Let Pw : V + V be a projection onto a subspace W and let A e Mnn (R) be the matrix of the linear map Pw with respect to a basis B = {ui, U2, ..., un} in V. Is it true that A2 = A? Give explanations to your answer.

Answer :

Final answer:

The formula for the projection of a vector x onto the subspace W is projW(x) = (x • u1)u1 + (x • u2)u2.

Explanation:

When projecting a vector x onto a subspace W, we can use the formula:

projW(x) = (x • u1)u1 + (x • u2)u2

where u1 and u2 are the orthonormal basis vectors of the subspace W.

In this case, the subspace W is spanned by the vectors u1 and u2, which are given as {ui, u2}. The formula for the projection of a vector x onto W is obtained by taking the dot product of x with each basis vector and multiplying it by the corresponding basis vector. Finally, we add these two projections together to get the projection of x onto W.

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