The column space of
is the plane
. The closest vector
to
is the orthogonal projection of
onto
. That is,
where
is a normal vector to
. For example,
since the constant
does not matter, we instead define
.
Therefore,
.
20 3. Let 1 = 2 and = 5. Let W = Span{11, 13). (a) Give a geometric description of W. (b) Use the Gram-Schmidt process to find an orthogonal basis for W. (c) Let = 2 Find the closest point to į in W. (a) Use your orthogonal basis in part (b) to find an orthonormal basis for W.
1) Find the rank of A
2) Find the dimensions of Nul(A) and Col(A)
3) How do the dimensions of Nul(A) and Col(A) relate to the
number of columns of A ?
9 3 2 27 18 A 6 9 2 2 Question 4. (15 pts) Let the matrix A be the same as in Question 3. (1). Find the rank of A. (2). Find the dimensions of Nul(A) and Col(.A). (3). How do the dimensions of Nul(A) and Col(A)...
3 5 1 -1 -3 1 1. Let A b 1C Col(A) 0 2 1 5 2 1 5 8 a) Find an orthonormal basis for C b) Find a QR factorization for A. c) Find b projc(b). Does b= b? What does this tell us about the number of solutions to Ax = b?
Problem 9. (1 point) T -5 10 1 Let A= and w= 2 -4 Is w in Col(A)? Type "yes" or "no". Is w in Nul(A)? Type "yes" or "no". Note: You can earn partial credit on this problem.
1 2 -3 1 -6 -2 5 2. 4. (10 points) Let A = (a) (5 points) Find a basis for col(A) and calculate rank(A). (b) (5 points) Find a basis for null(A) and calculate nullity(A).
5 1 -2 0-4 Let A=0 0 0 0 13 1 -2 0 -3 5 a. Find a basis for Col A and find Rank A. b. Find a basis for Nul A.
(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.
4. Let L: V→ W be a linear map. Let w be an element of W. Let uo be an ele- ment of V such that LvO-w. Show that any solution of the equation L(X)-w is of type uo + u, where u is an element of the kernel of L.
Let u = [1, 3, -2], v = [-1, 1, 1], w = [5, 1, 4]. a) Check if the system of vectors {u,v,w} is an orthogonal or othonormal basis of E3. b) Find the coordinates of the vector [1,0,1] in this basis.
Find a basis for Col(A) and a basis for Nul(A)
Question 3. (20 pts) Let A= 3 9-27 2-6 18 3 9 -2 2 Find a basis for Col(A) and a basis for Nul(A).