1 and w2 Write the vector as w +u, with w in W andu in W. 2.

suppose that x↓1,..,x↓(n-1) are linearly independent vectors in R^n and also x↓n is notin the span of x↓1,..,x↓(n-1) . 1. prove that x↓1,...,x↓n is a basis of R^n. 2. Prove that if x↓(n+1)∈R^n, then x↓1,..,x↓n,x↓n+1 are not linearly independent

Q1) Let S (x-3, x² + 2x, x² + 1). Is S is span (P2(x)) or not? Q2) Let T= ((2,-2,4), (3,-5,4), (0,1,1)). Show that whether T is linearly independent or not? Q3) Let N = (vv) be a basis for vector space V Show that every vector in V can be written as uniquely linear combination of vectors of V.

4. Consider the subspace V of R' given by 1 3 V = span (a) Find an orthonormal basis for the orthogonal complement V-. (b) Find the closest point to (1, 1, 1, 1) in V+.