Let V = M2(R), and let U be the span of
S =
... Let v = , u = , and let W the subspace of R4 spanned by v and u. Find a basis of W .
2 points) Let H be the subspace of P2 spanned by 2x2 - 6x +3, x2 -2x 1 and -2r221 (a) A basis for H is Enter a polynomial or a list of polynomials separated by commas, in terms of lower-case x . For example x+1,x-2 (b) The dimension of H is c)Is (2x2 6x +3, x2 - 2x +1, -2x2 +2x 1 a basis for P2? 2 points) Let H be the subspace of P2 spanned by 2x2 -...
(1) Let u = (-1,2) and v = (3, 1). (a) (5] Find graphically the vector w = (2u - v). (b) (5] Find algebraically the vector z=3u - 2 (2) (a) [5] Write u ='(1, -5, -1) as a linear combination of v1 = (1,2,0), v2 = (0,1,-1), V3 = (2,1,1). (b) (5] Are the 4 vectors u, V1, V2, V3 linearly independent? Explain your answer. (C) (5) Are the 2 vectors V, V3 linearly independent? Explain your answer....
Q6. Let W be the subspace of R' spanned by the vectors u. = 3(1, -1,1,1), uz = 5(–1,1,1,1). (a) Check that {uj,uz) is an orthonormal set using the dot product on R. (Hence it forms an orthonormal basis for W.) (b) Let w = (-1,1,5,5) EW. Using the formula in the box above, express was a linear combination of u and u. (c) Let v = (-1,1,3,5) = R'. Find the orthogonal projection of v onto W.
(a) Suppose that ū,ū e R". Show u2u-22||2 2해2 (b) (The Pythagoras Theorem) Suppose that u, v e R". Show that ul if and only if ||ü + 해2 (c) Let W be a subspace of R" with an orthogonal basis {w1, ..., w,} and let {ö1, ..., ūg} 22 orthogonal basis for W- (i) Explain why{w1, ..., üp, T1, .., T,} is an (ii Explain why the set in (i) spans R". (iii Show that dim(W) + dim(W1) be...
Problem 4. Let V be the vector space of all infinitely differentiable functions f: [0, ] -» R, equipped with the inner product f(t)g(t)d (f,g) = (a) Let UC V be the subspace spanned by B = (sinr, cos x, 1) (you may assume without proof that B is linearly independent, and hence a basis for U). Find the B-matrix [D]93 of the "derivative linear transformation" D : U -> U given by D(f) = f'. (b) Let WC V...
1. Let u - (1,1,2), v = (1,2,1), and w (2,1,1) in R. and consider • the parallelogram B = {s(3v) + t-w) 0 <s,t<1, s,te R} spanned/formed by the vectors (3v) and (-w), and • the parallelepiped P = {ru + s(3v) + (-w) 0 <T,,t<1, r, s, t€ R} [10] spanned formed by vectors u. (3v). and (-w) We take the parallelogram B as a base of P. (a) Does the ordered vector triple (v xw, 3v, -w),...
Step by step for #8 1) Given (1 2 3 1 0 11 1 5 2 1 A= -2 -5 -4 -1 1 ( 3 5 11 4 1 Find the basis and dimension for the row, the column spaces, and the null space NA Also, state the rank, the nullity of A 2) The subspace of W in R spanned by vectors u =(2.-2.1) v =(1,2,2) is a plane passing thru the origin. Express the vector w=(1,0.2) in the...
(1 point) Let u = 1. VE L . and let W the subspace of R4 spanned by {u, v}. Find a basis for WI. Answer:
Let P2 be the vector space of all polynomials of degree 2 or less, and let H be the subspace spanned by 8x−5x2+3, 2x-2x2+1 and 3x2-1. a) The dimensions of the subspace H is ___________? b) Is {8x-5x2+3, 2x-2x2+1, 3x2-1} a basis for P2? ________(be sure to explain and justify answer) c) A basis for the subspace H is {_________}? enter a polynomial or comma separated list of polynomials